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\begin{center}
\vskip 1cm{\LARGE\bf New Ramanujan-Type Formulas and \\
\vskip .1in 
Quasi-Fibonacci Numbers of Order 7}
\vskip 1cm
\large
Roman Witu{\l}a and Damian S{\l}ota \\
Institute of Mathematics \\
Silesian University of Technology \\
Kaszubska 23 \\
Gliwice 44-100 \\
Poland \\
\href{mailto:r.witula@polsl.pl}{\tt r.witula@polsl.pl} \\
\href{mailto:d.slota@polsl.pl}{\tt d.slota@polsl.pl} \\
\end{center}


\vskip .2in




\begin{abstract}
We give applications of the quasi-Fibonacci numbers of order $7$
and the so-called sine-Fibonacci numbers of order $7$
and many other new kinds of recurrent sequences to
the decompositions of some polynomials.
We also present the characteristic equations, generating functions
and some properties of all these sequences.
Finally, some new Ramanujan-type formulas are generated.
\end{abstract}





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\section{Introduction}\label{roz1}

The scope of the paper is the generalization of the following
decompositions of polynomials \cite{GrzymkowskiWitula,Ireland,Shevelev,surowski,Watkins}:
\begin{align}
\big( \mathbb{X}  - &2 \sin (\tfrac{2\,\pi}{7})\big)
\big( \mathbb{X}  - 2 \sin (\tfrac{4\,\pi}{7})\big)
\big( \mathbb{X}  - 2 \sin (\tfrac{8\,\pi}{7})\big)
=\mathbb{X}^3 -\sqrt{7}\, \mathbb{X}^2 +\sqrt{7},\label{w1-a}\\
\big( \mathbb{X}  - &4 \sin^2 (\tfrac{2\,\pi}{7})\big)
\big( \mathbb{X}  - 4 \sin^2 (\tfrac{4\,\pi}{7})\big)
\big( \mathbb{X}  - 4 \sin^2 (\tfrac{8\,\pi}{7})\big)
=\mathbb{X}^3 - 7\, \mathbb{X}^2 + 14\, \mathbb{X} - 7,\label{w1-b}\\
\big( \mathbb{X}  - &8 \sin^3 (\tfrac{2\,\pi}{7})\big)
\big( \mathbb{X}  - 8 \sin^3 (\tfrac{4\,\pi}{7})\big)
\big( \mathbb{X}  - 8 \sin^3 (\tfrac{8\,\pi}{7})\big)
=\mathbb{X}^3-4\, \sqrt{7}\, \mathbb{X}^2 +21\, \mathbb{X} +7\, \sqrt{7} ,\label{w1-c}\\
\big( \mathbb{X}  - &2 \cos (\tfrac{2\,\pi}{7})\big)
\big( \mathbb{X}  - 2 \cos (\tfrac{4\,\pi}{7})\big)
\big( \mathbb{X}  - 2 \cos (\tfrac{8\,\pi}{7})\big)
=\mathbb{X}^3 + \mathbb{X}^2 - 2\, \mathbb{X} -1,\label{w1-d}\\
\big( \mathbb{X}  - &4 \cos^2 (\tfrac{2\,\pi}{7})\big)
\big( \mathbb{X}  - 4 \cos^2 (\tfrac{4\,\pi}{7})\big)
\big( \mathbb{X}  - 4 \cos^2 (\tfrac{8\,\pi}{7})\big)
=\mathbb{X}^3 - 5\, \mathbb{X}^2 + 6\, \mathbb{X} - 1,\label{w1-e}\\
\big( \mathbb{X}  - &8 \cos^3 (\tfrac{2\,\pi}{7})\big)
\big( \mathbb{X}  - 8 \cos^3 (\tfrac{4\,\pi}{7})\big)
\big( \mathbb{X}  - 8 \cos^3 (\tfrac{8\,\pi}{7})\big)
=\mathbb{X}^3+4\, \mathbb{X}^2-11\, \mathbb{X}-1,\label{w1-f}\\
\big( \mathbb{X}  - &8 \sin (\tfrac{2\,\pi}{7})\, \cos^2 (\tfrac{8\,\pi}{7})\big)
\big( \mathbb{X}  - 8 \sin (\tfrac{4\,\pi}{7})\, \cos^2 (\tfrac{2\,\pi}{7})\big)
\big( \mathbb{X}  - 8 \sin (\tfrac{8\,\pi}{7})\, \cos^2 (\tfrac{4\,\pi}{7})\big)=\nonumber\\
&=\mathbb{X}^3 - 3\, \sqrt{7}\, \mathbb{X}^2 + 14\, \mathbb{X} + \sqrt{7},\label{w1-g}\\
etc.& \nonumber
\end{align}
The main incentive for generating the decompositions
of these polynomials
is provided by the properties
of the so-called quasi-Fibonacci numbers of order $7$,
$A_{n}(\delta)$,
$B_{n}(\delta)$ and $C_{n}(\delta)$, $n\in \mathbb{N}$, described in~\cite{wsw1}
by means of the relations
\begin{equation}
(1+\delta\, (\xi^{k}+\xi^{6k}))^n =
A_n(\delta)+B_n(\delta)\, (\xi^{k}+\xi^{6k})+C_n(\delta)\, (\xi^{2k}+\xi^{5k})
\end{equation}
for $k=1,2,3$,
where $\xi\in\mathbb{C}$
is a~primitive root of unity of order $7$ (i.e.,~$\xi^7=1$ and $\xi\neq 1$),
$\delta\in \mathbb{C}$, $\delta\neq 0$.
Besides, an essential r{\^{o}}le in the decompositions of polynomials
discussed in the paper is played by related numbers
($\delta\in \mathbb{C}$, $n\in\mathbb{N}$):
\begin{multline}\label{w-and}
\mathcal{A}_{n}(\delta) := 3\, A_{n}(\delta) - B_{n}(\delta)
-C_{n}(\delta) = \\
=\big(1+\delta\, (\xi+\xi^6)\big)^n
+\big(1+\delta\, (\xi^2+\xi^5)\big)^n
+\big(1+\delta\, (\xi^3+\xi^4)\big)^n
\end{multline}
and
\begin{align}
\mathcal{B}_{n}(\delta) &:= \frac{1}{2} \big( \big(\mathcal{A}_{n}(\delta)\big)^{2} -
\mathcal{A}_{2n}(\delta) \big) = \nonumber\\
&=\Big( \big( 1 + 2\,\delta\, \cos (\tfrac{2\, \pi}{7}) \big)
\big( 1 + 2\,\delta\, \cos (\tfrac{4\, \pi}{7}) \big)\Big)^{n}+{}\nonumber\\
&\phantom{=}\hspace*{2ex} {} +
\Big( \big( 1 + 2\,\delta\, \cos (\tfrac{2\, \pi}{7}) \big)
\big( 1 + 2\,\delta\, \cos (\tfrac{8\, \pi}{7}) \big)\Big)^{n}+{}\nonumber\\
&\phantom{=}\hspace*{2ex}  {} +
\Big( \big( 1 + 2\,\delta\, \cos (\tfrac{4\, \pi}{7}) \big)
\big( 1 + 2\,\delta\, \cos (\tfrac{8\, \pi}{7}) \big)\Big)^{n}={}\nonumber\\
&= 3\,\big(A_{n}(\delta)\big)^{2}
- 2\, A_{n}(\delta)\, B_{n}(\delta)
- 2\, A_{n}(\delta)\, C_{n}(\delta) + {} \nonumber \\
&\phantom{=}\hspace*{2ex}  {} + 3\, B_{n}(\delta)\, C_{n}(\delta)
-2\,\big(B_{n}(\delta)\big)^{2}
-2\,\big(C_{n}(\delta)\big)^{2} = {} \nonumber\\
&=A_{n}(\delta)\, \big( 2\, \mathcal{A}_{n}(\delta) - 3 A_{n}(\delta) \Big)
- 2\, \big( B_{n}(\delta) - C_{n}(\delta) \Big)^{2}
- B_{n}(\delta)\, C_{n}(\delta).\label{w-bnd}
\end{align}
Furthermore, to simplify the formulas, we will write
\begin{equation}
\mathcal{A}_{n}=\mathcal{A}_{n}(1),\ \
\mathcal{B}_{n}=\mathcal{B}_{n}(1),\ \
A_{n}=A_{n}(1),\ \
B_{n}=B_{n}(1)\ \  \mbox{and}\ \
C_{n}=C_{n}(1),
\end{equation}
for every $n\in \mathbb{N}$.
We note that the tables of values of these numbers can be found in the article~\cite{wsw1}.

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\section{Basic decompositions}\label{roz2a}

Witu{\l}a et al.\ \cite{wsw1} determined the following two formulas:
\begin{multline}
\big( \mathbb{X} - \big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - \big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - \big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} = \mathbb{X}^3 - \mathcal{B}_{n}\, \mathbb{X}^2
+ (-1)^{n}\, \mathcal{A}_{n} \, \mathbb{X} - 1\label{wzor-gw1new}
\end{multline}
and
\begin{multline}\label{w1.10}
\big( \mathbb{X} - \big( 1+ 2\,\delta\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - \big( 1+ 2\,\delta\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - \big( 1+ 2\,\delta\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} = \mathbb{X}^3 - \mathcal{A}_{n} (\delta)\, \mathbb{X}^2
+ \mathcal{B}_{n}(\delta) \, \mathbb{X} - \big(1-\delta-2\, \delta^2+\delta^3\big)^n.
\end{multline}
From~(\ref{w1.10}) three special formulas follow:
\begin{multline}\label{wiel-114}
\big( \mathbb{X} - \big( 1+ \cos^2 (\tfrac{2\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - \big( 1+ \cos^2 (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - \big( 1+ \cos^2 (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} = \mathbb{X}^3 - \big( \tfrac{3}{2}\big)^{n}\,
\mathcal{A}_{n} \big(\tfrac{1}{6}\big)\, \mathbb{X}^2
+ \big( \tfrac{3}{2}\big)^{2n}\,  \mathcal{B}_{n} \big(\tfrac{1}{6}\big)\, \mathbb{X}
- \big( \tfrac{13}{8}\big)^{2n},
\end{multline}
\begin{multline}
\big( \mathbb{X} - \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{2n} \big)
\big( \mathbb{X} - \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{2n} \big)
\big( \mathbb{X} - \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{2n} \big) = {}\\
{} = \mathbb{X}^3 - 2^{n}\,  \mathcal{A}_{n} \big(-\tfrac{1}{2}\big)\, \mathbb{X}^2
+ 2^{2n}\,  \mathcal{B}_{n} \big(-\tfrac{1}{2}\big)\, \mathbb{X} - 7^n
\end{multline}
and
\begin{multline}\label{wz116}
\big( \mathbb{X} - \big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{2n} \big)
\big( \mathbb{X} - \big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{2n} \big)
\big( \mathbb{X} - \big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n} \big) = {}\\
{} = \mathbb{X}^3 - 2^{n}\,  \mathcal{A}_{n} \big(\tfrac{1}{2}\big)\, \mathbb{X}^2
+ 2^{2n}\,  \mathcal{B}_{n} \big(\tfrac{1}{2}\big)\, \mathbb{X} - 1.
\end{multline}
Comparing formulas~(\ref{wzor-gw1new}) and~(\ref{wz116}) two new identities
are generated
\begin{equation}
\mathcal{A}_{2n}=2^{2n}\,  \mathcal{B}_{n} \big(\tfrac{1}{2}\big)
\quad \mbox{ and } \quad
\mathcal{B}_{2n}=2^{n}\,  \mathcal{A}_{n} \big(\tfrac{1}{2}\big).
\end{equation}


We also have the decomposition
\begin{multline}\label{w1.19}
\Big( \mathbb{X} - \big( \big(1+\delta\, (\xi+\xi^6)\big)\,
\big(1+\delta\, (\xi^2+\xi^5)\big)\big)^{n} \Big)\,
\Big( \mathbb{X} - \big( \big(1+\delta\, (\xi+\xi^6)\big)\,
\big(1+\delta\, (\xi^3+\xi^4)\big)\big)^{n} \Big)\times{}\\
{}\times \Big( \mathbb{X} - \big( \big(1+\delta\, (\xi^2+\xi^5)\big)\,
\big(1+\delta\, (\xi^3+\xi^4)\big)\big)^{n} \Big) = {}\\
{} =
\Big( \mathbb{X} - \big( \big(1+2\,\delta\, \cos (\tfrac{2\, \pi}{7} ) \big)\,
\big(1+2\,\delta\, \cos (\tfrac{4\, \pi}{7} ) \big) \big)^{n} \Big)\,
\Big( \mathbb{X} - \big( \big(1+2\,\delta\, \cos (\tfrac{2\, \pi}{7} ) \big)\,
\big(1+2\,\delta\, \cos (\tfrac{6\, \pi}{7} ) \big) \big)^{n} \Big)\times {} \\
{}\times
\Big( \mathbb{X} - \big( \big(1+2\,\delta\, \cos (\tfrac{4\, \pi}{7} ) \big)\,
\big(1+2\,\delta\, \cos (\tfrac{6\, \pi}{7} ) \big) \big)^{n} \Big)
={}\\
{}=
\mathbb{X}^{3} - \mathcal{B}_{n}(\delta)\, \mathbb{X}^{2} +
\big(1-\delta-2\, \delta^2+\delta^3\big)\, \mathcal{A}_{n}(\delta)\, \mathbb{X} -
\big(1-\delta-2\, \delta^2+\delta^3\big)^{2n} = {} \\
{} := r_{n}(\mathbb{X}; \delta).
\end{multline}

Now let us set
\begin{align}
\Xi_{n} &:= \Xi_{n} (\delta, \varepsilon, \eta ) =
2^n\, \Big( \delta\, \cos^n \big(\tfrac{2\, \pi}{7}\big) +
\varepsilon\, \cos^n \big(\tfrac{4\, \pi}{7}\big) +
\eta\, \cos^n \big(\tfrac{8\, \pi}{7}\big) \Big),\label{w2.8}\\
\Upsilon_{n} &:= \Upsilon_{n} (\delta, \varepsilon, \eta ) =
2^n\, \Big( \varepsilon\, \cos^n \big(\tfrac{2\, \pi}{7}\big) +
\eta\, \cos^n \big(\tfrac{4\, \pi}{7}\big) +
\delta\, \cos^n \big(\tfrac{8\, \pi}{7}\big) \Big),\\
\Theta_{n} &:= \Theta_{n} (\delta, \varepsilon, \eta ) =
2^n\, \Big( \eta\, \cos^n \big(\tfrac{2\, \pi}{7}\big) +
\delta\, \cos^n \big(\tfrac{4\, \pi}{7}\big) +
\varepsilon\, \cos^n \big(\tfrac{8\, \pi}{7}\big) \Big)
\end{align}
for any $\delta, \varepsilon, \eta \in \mathbb{C}$ and $n\in \mathbb{N}_0$.


\begin{lemma}\label{lem2.1}
The following general decomposition formula holds
\begin{multline}\label{choinka}
(\mathbb{X} -\Xi_{n})
(\mathbb{X} -\Upsilon_{n})
(\mathbb{X} -\Theta_{n}) =\\
= \mathbb{X}^3 - (\delta+\varepsilon+\eta)\, \mathcal{A}_{n}\, \mathbb{X}^2
+\tfrac{1}{2}\, (-1)^n\, \big( (\delta+\varepsilon)^2 +(\delta+\eta)^2
+(\varepsilon+\eta)^2 \big)\, \mathcal{A}_{n}\, \mathbb{X} - {}\\
{} -
\Big[
\delta\,\varepsilon\,\eta\, (\mathcal{B}_{3n}+3) + \delta^3 + \varepsilon^3 + \eta^3
+(-1)^n\, \big( \delta\,\varepsilon^2 + \varepsilon\, \eta^2  + \eta\, \delta^2 \big)\,
\mathcal{A}_{n}(-1) + {}\\
{}+ 2^{2n}\, \big( \delta^2\,\varepsilon + \varepsilon^2\, \eta  + \eta^2\, \delta \big)\,
\sum_{k=0}^{n} (-1)^k\, \binom{n}{k}\, \mathcal{A}_{2n-k} \big(\tfrac{1}{2}\big)
\Big].
\end{multline}
\end{lemma}

Below an illustrative example connected with Lemma~\ref{lem2.1} will be presented.

\begin{example}\label{example2.2}
{\rm
A.~M.~Yaglom and I.~M.~Yaglom~\cite{Yaglom}
(see also~\cite[problem~230]{ShklyarskyUSA} and~\cite[problem~329]{ShklyarskyMir})
considered the following polynomial:
$$
w(x)=\binom{2m + 1}{1}\, x^m
- \binom{2m + 1}{3}\, x^{m-1} +
\binom{2m + 1}{5}\, x^{m-2} - \ldots
$$
and proved that it has the roots
$$
x_k = \cot^2 \Big( \frac{k \, \pi}{2m + 1} \Big),
\qquad k = 1, 2, \ldots, m.
$$
In particular, for $m = 3$ taking into account that
$$
\cot^2 \big( \tfrac{\pi}{7} \big) =
\cot^2 \big( \tfrac{8\, \pi}{7} \big),
\qquad
\cot^2 \big( \tfrac{3\,\pi}{7} \big) =
\cot^2 \big( \tfrac{4\, \pi}{7} \big)
$$
we have the decomposition (see also formula~(\ref{w6.14}) below):
$$
\big( x - \cot^2 \big( \tfrac{2\,\pi}{7} \big) \big) \,
\big( x - \cot^2 \big( \tfrac{4\,\pi}{7} \big) \big) \,
\big( x - \cot^2 \big( \tfrac{8\,\pi}{7} \big) \big) =
x^3 - 5\, x^2 + 3\, x - \tfrac{1}{7}
$$
and as a corollary~-- the decomposition
\begin{equation}\label{ex-w5}
\big( x + 7\, \cot^2 \big( \tfrac{2\,\pi}{7} \big) \big) \,
\big( x + 7\, \cot^2 \big( \tfrac{4\,\pi}{7} \big) \big) \,
\big( x + 7\, \cot^2 \big( \tfrac{8\,\pi}{7} \big) \big) =
x^3 + 35\, x^2 + 147\, x +49.
\end{equation}
According to~(\ref{w2.8}) for $n = 1$, let us try to find a~linear combination
$$
-7\, \cot^2 \big( \tfrac{2\,\pi}{7} \big) =
2\,\Big( \delta\, \cos \big(\tfrac{2\, \pi}{7}\big) +
\varepsilon\, \cos \big(\tfrac{4\, \pi}{7}\big) +
\eta\, \cos \big(\tfrac{8\, \pi}{7}\big) \Big)
$$
or, the same,
\begin{multline*}
-7\, \cos^2 \big( \tfrac{2\,\pi}{7} \big) =
2\,\Big( \delta\, \cos \big(\tfrac{2\, \pi}{7}\big) +
\varepsilon\, \cos \big(\tfrac{4\, \pi}{7}\big) +
\eta\, \cos \big(\tfrac{8\, \pi}{7}\big) \Big) - {}\\
{}- \delta\, \cos^3 \big(\tfrac{2\, \pi}{7}\big)
-\varepsilon\,
\cos \big(\tfrac{4\, \pi}{7}\big)\,
\cos^2 \big(\tfrac{2\, \pi}{7}\big)
-\eta\,
\cos \big(\tfrac{8\, \pi}{7}\big)\,
\cos^2 \big(\tfrac{2\, \pi}{7}\big).
\end{multline*}
Decreasing powers and taking into account the identity
$$
\cos \big(\tfrac{2\, \pi}{7}\big) +
\cos \big(\tfrac{4\, \pi}{7}\big) +
\cos \big(\tfrac{8\, \pi}{7}\big) = -\frac{1}{2}
$$
we find
$$
\big( 2\, \delta+ \varepsilon -3\, \eta \big)
\cos \big(\tfrac{2\, \pi}{7}\big) +
\big( \delta+ 3\,\varepsilon -3\, \eta +7 \big)
\cos \big(\tfrac{4\, \pi}{7}\big) =
-\frac{\delta}{2} +\frac{\varepsilon}{2} +\eta-7.
$$
Thus, for finding $\delta$, $\varepsilon$, $\eta$ we have the linear system
(see Corollary~2.5 in~\cite{wsw1}):
$$
\left\{
\begin{array}{l}
2\, \delta+ \varepsilon -3\, \eta = 0 \\
\delta+ 3\,\varepsilon -3\, \eta = -7 \\
\delta - \varepsilon - 2\, \eta  = -14
\end{array}
\right.
$$
with the solution
$$
\delta=17,\qquad
\varepsilon=5,\qquad
\eta=13.
$$
Hence,
$$
-7\, \cot^2 \big( \tfrac{2\,\pi}{7} \big) =
2\,\Big( 17\, \cos \big(\tfrac{2\, \pi}{7}\big) +
5\, \cos \big(\tfrac{4\, \pi}{7}\big) +
13\, \cos \big(\tfrac{8\, \pi}{7}\big) \Big).
$$
Analogously, we obtain
\begin{align*}
-7\, \cot^2 \big( \tfrac{4\,\pi}{7} \big) &=
2\,\Big( 13\, \cos \big(\tfrac{2\, \pi}{7}\big) +
17\, \cos \big(\tfrac{4\, \pi}{7}\big) +
5\, \cos \big(\tfrac{8\, \pi}{7}\big) \Big),\\
-7\, \cot^2 \big( \tfrac{8\,\pi}{7} \big) &=
2\,\Big( 5\, \cos \big(\tfrac{2\, \pi}{7}\big) +
13\, \cos \big(\tfrac{4\, \pi}{7}\big) +
17\, \cos \big(\tfrac{8\, \pi}{7}\big) \Big).
\end{align*}
Thus the decomposition~(\ref{ex-w5}) corresponds to Lemma~\ref{lem2.1} with
$$
\Xi_{1}(17, 5, 13),\quad
\Upsilon_{1}(17, 5, 13),\quad
\Theta_{1}(17, 5, 13).
$$
}
\end{example}

\begin{remark}
{\rm
The identity~(\ref{ex-w5}) was found earlier by Shevelev~\cite{Shevelev}.
}
\end{remark}

\begin{remark}
{\rm
The formula~(\ref{choinka}), in some cases which are subject of our interest,
especially when coefficients  $\delta$, $\varepsilon$, $\eta$ are the corresponding
values of trigonometric functions, becomes rather complicated.
Thus, in the next two sections we attempt to designate the relevant
coefficients of decomposition~(\ref{choinka}),
including the recurrent coefficients, on the grounds of new sequences
that are easier to analyze.
}
\end{remark}







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\section{The first group of special cases of~(\ref{choinka})}

Let us set
\begin{align}
a_{n} &= 2^{2n+1}\, \Big[
\sin (\tfrac{2\, \pi}{7})
\big( \cos (\tfrac{8\, \pi}{7}) \big)^{2n} +
\sin (\tfrac{4\, \pi}{7})
\big( \cos (\tfrac{2\, \pi}{7}) \big)^{2n} +
\sin (\tfrac{8\, \pi}{7})
\big( \cos (\tfrac{4\, \pi}{7}) \big)^{2n} \Big],\label{w2.1}\\
b_{n} &= 2^{2n+1}\, \Big[
\sin (\tfrac{4\, \pi}{7})
\big( \cos (\tfrac{8\, \pi}{7}) \big)^{2n} +
\sin (\tfrac{8\, \pi}{7})
\big( \cos (\tfrac{2\, \pi}{7}) \big)^{2n} +
\sin (\tfrac{2\, \pi}{7})
\big( \cos (\tfrac{4\, \pi}{7}) \big)^{2n} \Big],\label{w2.2}\\
c_{n} &= 2^{2n+1}\, \Big[
\sin (\tfrac{8\, \pi}{7})
\big( \cos (\tfrac{8\, \pi}{7}) \big)^{2n} +
\sin (\tfrac{2\, \pi}{7})
\big( \cos (\tfrac{2\, \pi}{7}) \big)^{2n} +
\sin (\tfrac{4\, \pi}{7})
\big( \cos (\tfrac{4\, \pi}{7}) \big)^{2n} \Big],\label{w2.3}
\end{align}
for $n=0,1,2,\ldots$


\begin{lemma}\label{f2s-lem1}
The following recurrence relations hold:
\begin{equation}\label{f2s-w1g}
\left\{
\begin{array}{l}
a_{n+1} = 2\, a_{n} + b_{n},\\
b_{n+1} = a_{n} + 2\, b_{n} - c_{n},\\
c_{n+1} = c_{n} - b_{n},\\
\end{array}
\right.
\end{equation}
for $n=0,1,2,\ldots$ and $a_{0}=b_{0}=c_{0}=\sqrt{7}$.
Moreover, elements of each sequences $\{a_n\}_{n=0}^{\infty}$,
$\{b_n\}_{n=0}^{\infty}$ and $\{c_n\}_{n=0}^{\infty}$
satisfy the recurrence equation
\begin{equation}\label{f2s-w2g}
x_{n+2} - 5\, x_{n+1} + 6\, x_{n} - x_{n-1} = 0,
\qquad n=0,1,2,\ldots
\end{equation}
(in view of decomposition~(\ref{w1-e}) an appropriate characteristic polynomial
is compatible with the definition of numbers $a_n$, $b_n$ and~$c_n$).
\end{lemma}

The first twelve values of numbers $a_{n}^{*}=a_{n}/\sqrt{7}$,
$b_{n}^{*}=b_{n}/\sqrt{7}$ and $c_{n}^{*}=c_{n}/\sqrt{7}$ are presented
in Table~\ref{f2s-tab1}.



Now, let us set
\begin{align}
\alpha_{n} &= 2^{2n}\, \Big[
\sin (\tfrac{2\, \pi}{7})
\big( \cos (\tfrac{8\, \pi}{7}) \big)^{2n-1} +
\sin (\tfrac{4\, \pi}{7})
\big( \cos (\tfrac{2\, \pi}{7}) \big)^{2n-1} +
\sin (\tfrac{8\, \pi}{7})
\big( \cos (\tfrac{4\, \pi}{7}) \big)^{2n-1} \Big],\\
\beta_{n} &= 2^{2n}\, \Big[
\sin (\tfrac{4\, \pi}{7})
\big( \cos (\tfrac{8\, \pi}{7}) \big)^{2n-1} +
\sin (\tfrac{8\, \pi}{7})
\big( \cos (\tfrac{2\, \pi}{7}) \big)^{2n-1} +
\sin (\tfrac{2\, \pi}{7})
\big( \cos (\tfrac{4\, \pi}{7}) \big)^{2n-1} \Big],\\
\gamma_{n} &= 2^{2n}\, \Big[
\sin (\tfrac{8\, \pi}{7})
\big( \cos (\tfrac{8\, \pi}{7}) \big)^{2n-1} +
\sin (\tfrac{2\, \pi}{7})
\big( \cos (\tfrac{2\, \pi}{7}) \big)^{2n-1} +
\sin (\tfrac{4\, \pi}{7})
\big( \cos (\tfrac{4\, \pi}{7}) \big)^{2n-1} \Big],
\end{align}
for $n=1,2,\ldots$

\begin{lemma}\label{f2s-lem2}
We have
$$
\alpha_1=0,\qquad
\beta_1=-2\, \sqrt{7},\qquad \mbox{and} \qquad
\gamma_1=\sqrt{7}.
$$
The elements of sequences $\{\alpha_n\}_{n=1}^{\infty}$,
$\{\beta_n\}_{n=1}^{\infty}$ and $\{\gamma_n\}_{n=1}^{\infty}$
satisfy the system of recurrence relations~(\ref{f2s-w1g}) and, selectively,
recurrence relation~(\ref{f2s-w2g}).
\end{lemma}

The first twelve values of numbers $\alpha_{n}^{*}=\alpha_{n}/\sqrt{7}$,
$\beta_{n}^{*}=\beta_{n}/\sqrt{7}$ and $\gamma_{n}^{*}=\gamma_{n}/\sqrt{7}$
are presented in Table~\ref{f2s-tab1}.

\begin{remark}
{\rm
There exists a~simple relationships between
numbers $\alpha_n$, $\beta_n$, $\gamma_n$, $n\in \mathbb{N}$,
and $a_n$, $b_n$, $c_n$, $n\in \mathbb{N}$.
We have
$$
\alpha_{n} \equiv c_n,\qquad
\beta_{n}  \equiv {-}a_{n-1} - b_{n-1} ,\qquad
\gamma_{n} \equiv a_{n-1}.
$$
These relations are easily derived from the definitions of respective numbers,
for example
\begin{align*}
\beta_{n} &= 2^{2(n-1)+1} \Big[
2\, \sin (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7})
\big( \cos (\tfrac{8\, \pi}{7}) \big)^{2(n-1)}+{}\\
&\phantom{=}\hspace*{2ex} {}+
2\, \sin (\tfrac{8\, \pi}{7})\, \cos (\tfrac{2\, \pi}{7})
\big( \cos (\tfrac{2\, \pi}{7}) \big)^{2(n-1)}+%{}\\
%&\phantom{=}\hspace*{2ex} {}+
2\, \sin (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7})
\big( \cos (\tfrac{4\, \pi}{7}) \big)^{2(n-1)}
\Big]= \\
&=
2^{2(n-1)+1} \Big[
-\big(\sin (\tfrac{2\, \pi}{7}) + \sin (\tfrac{4\, \pi}{7})\big)
\big( \cos (\tfrac{8\, \pi}{7}) \big)^{2(n-1)}+{}\\
&\phantom{=}\hspace*{2ex} {} +
\big({-}\sin (\tfrac{4\, \pi}{7}) - \sin (\tfrac{8\, \pi}{7})\big)
\big( \cos (\tfrac{2\, \pi}{7}) \big)^{2(n-1)}-%{}\\
%&\phantom{=}\hspace*{2ex} {} -
\big( \sin (\tfrac{2\, \pi}{7}) + \sin (\tfrac{8\, \pi}{7}) \big)
\big( \cos (\tfrac{4\, \pi}{7}) \big)^{2(n-1)}
\Big]= {}\\
&= - a_{n-1} - b_{n-1}. \\
\end{align*}
}
\end{remark}


Let us also set
\begin{align}
f_{n} &= 2^{n+1}\, \Big[
\cos (\tfrac{2\, \pi}{7})
\big( \cos (\tfrac{4\, \pi}{7}) \big)^{n} +
\cos (\tfrac{4\, \pi}{7})
\big( \cos (\tfrac{8\, \pi}{7}) \big)^{n} +
\cos (\tfrac{8\, \pi}{7})
\big( \cos (\tfrac{2\, \pi}{7}) \big)^{n} \Big],\label{w2.11a}\\
g_{n} &= 2^{n+1}\, \Big[
\cos (\tfrac{8\, \pi}{7})
\big( \cos (\tfrac{4\, \pi}{7}) \big)^{n} +
\cos (\tfrac{2\, \pi}{7})
\big( \cos (\tfrac{8\, \pi}{7}) \big)^{n} +
\cos (\tfrac{4\, \pi}{7})
\big( \cos (\tfrac{2\, \pi}{7}) \big)^{n} \Big],\label{w2.11b}\\
h_{n} &= 2^{n+1}\, \Big[
\big( \cos (\tfrac{2\, \pi}{7}) \big)^{n+1} +
\big( \cos (\tfrac{4\, \pi}{7}) \big)^{n+1} +
\big( \cos (\tfrac{8\, \pi}{7}) \big)^{n+1} \Big],\label{w2.11}
\end{align}
for $n=0,1,2,\ldots$

\begin{lemma}\label{f2s-lem3}
We have
$$
f_0=g_0=h_0=-1,\qquad \mbox{and} \qquad
h_1=5
$$
and
\begin{equation}\label{f2s-w3g}
\left\{
\begin{array}{ll}
f_{n+1} = f_{n} + g_{n},& \quad n \geq 0,\\
g_{n+1} = f_{n} + h_{n},& \quad n \geq 0,\\
h_{n+1} = g_{n} + 2\, h_{n-1},& \quad n \geq 1.\\
\end{array}
\right.
\end{equation}
The elements of sequences $\{f_n\}_{n=0}^{\infty}$,
$\{g_n\}_{n=0}^{\infty}$ and $\{h_n\}_{n=0}^{\infty}$
satisfy the following recurrence relation (see formula~(\ref{w1-d})):
\begin{equation}\label{ww3}
x_{n+3}+x_{n+2}-2\, x_{n+1} - x_{n} =0,\qquad
n=0,1,\ldots
\end{equation}
\end{lemma}


%Proof
\begin{proof}
By~(\ref{f2s-w3g}) we obtain
\begin{align}
g_{n} & = f_{n+1}-f_{n},\label{ww1}\\
h_{n} & = g_{n+1}-f_{n} = f_{n+2}-f_{n+1}-f_{n},\label{ww2}
\end{align}
and, finally, the following identity
$$
f_{n+3}-f_{n+2}-f_{n+1} = 2\, \big( f_{n+1} - f_{n} - f_{n-1} \big) + f_{n+1} - f_{n},
$$
i.e.,
\begin{equation}\label{ww4}
f_{n+3}-f_{n+2}-4\, f_{n+1} +3\, f_{n} + 2\, f_{n-1} = 0.
\end{equation}
But we also have the following decomposition of respective characteristic polynomial
$$
x^4-x^3-4\, x^2+3\, x+2 = (x-2)(x^3+x^2-2\, x-1),
$$
which implies the following form of~(\ref{ww4}):
\begin{equation}\label{ww5}
f_{n+3}+f_{n+2}-2\, f_{n+1} - f_{n} = 2\, \big( f_{n+2} + f_{n+1} - 2\, f_{n} - f_{n-1} \big).
\end{equation}
Since
$$
f_{3}+f_{2}-2\, f_{1} - f_{0}  = 0
$$
so, we obtain the required identity~(\ref{ww3})
from~(\ref{ww5}), (\ref{ww1}) and~(\ref{ww2}).
\end{proof}
\bigskip

The first twelve elements of the sequences $\{f_n\}_{n=0}^{\infty}$,
$\{g_n\}_{n=0}^{\infty}$ and $\{h_n\}_{n=0}^{\infty}$
are presented in Table~\ref{f2s-tab1}.

\begin{remark}
{\rm
We note, that
\begin{equation}
h_{n-1}=\mathcal{B}_{n} =
\frac{1}{2} \Big( \big(\mathcal{A}_{n}\big)^{2} -
\mathcal{A}_{2n} \Big),\qquad n=1,2,\ldots
\end{equation}
is an accelerator sequence for Catalan's constant (see~\cite{bradley}
and  \seqnum{A094648}~\cite{sloan}).
}
\end{remark}

The next lemma contains a~sequence of eight identities
and simultanously six newly
defined auxiliary sequences of real numbers
$\{\widetilde{A}_{n}\}$, $\{\widetilde{B}_{n}\}$, $\{\widetilde{C}_{n}\}$,
$\{F_{n}\}$, $\{G_{n}\}$ and $\{H_{n}\}$.

\begin{lemma}\label{lem2.6}
The following identities hold
%
%
\begin{multline}\label{lem2.6-w1}
4 \, \cos (\tfrac{2\, \pi}{7})\,
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n}
+ 4 \, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} + {} \\
{} + 4 \, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n}  = {}\\
{} =
f_{n}^{2} + g_{n}^{2} - h_{n}^{2} + h_{2n+1} -2\, h_{2n}
- 4\, h_{2n-1} -f_{2n} := 2\, F_{n},
\end{multline}
%
%
\begin{multline}\label{lem2.6-w2}
2 \, \cos (\tfrac{2\, \pi}{7})\,
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{n}
+ 2 \, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} + {} \\
{}+ 2 \, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n}  = {}\\
{} =
F_{n} + h_{n}^{2} - f_{n}^{2} - h_{2n+1} + 2\, h_{2n}
+ 2\, h_{2n-1}  := G_{n},
\end{multline}
%
%
\begin{multline}\label{lem2.6-w3}
2 \, \cos (\tfrac{2\, \pi}{7})\,
\big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n}
+ 2 \, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} + {} \\
{}+ 2 \, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{n}  = {}\\
{} = f_{n}^{2} - 2\, ( h_{2n} + h_{2n-1} ) - F_{n} =
h_{n}^{2} - h_{2n+1} - G_{n} := H_{n},
\end{multline}
%
%
\begin{equation}\label{lem2.6-w4}
g_{n}^{2} - 2\, h_{2n-1} - f_{2n} = (-1)^{n} \big( \mathcal{A}_{n} + \mathcal{A}_{n+1} - 7\, A_{n}\big)
=G_{n}+F_{n},
\end{equation}
%
%
\begin{equation}\label{lem2.6-w5}
f_{n}^{2} - 2\, h_{2n} - 2\, h_{2n-1} = (-1)^{n} \big( \mathcal{A}_{n} - \mathcal{A}_{n-2} \big)
=H_{n}+F_{n},
\end{equation}
%
%
\begin{multline}\label{lem2.6-w6}
2 \, \sin (\tfrac{2\, \pi}{7})\,
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{2n}
+ 2 \, \sin (\tfrac{4\, \pi}{7})
\big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n} + {} \\
{}+ 2 \, \sin (\tfrac{8\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n}  = {}\\
{} = a_{2n} -a_{n}\, h_{2n-1}  + \tfrac{\sqrt{7}}{2}\, ( h_{2n-1}^{2} - h_{4n-1} )
:= \widetilde{A}_{n},
\end{multline}
%
%
\begin{multline}\label{lem2.6-w7}
2 \, \sin (\tfrac{2\, \pi}{7})\,
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n}
+ 2 \, \sin (\tfrac{4\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{2n} + {} \\
{}+ 2 \, \sin (\tfrac{8\, \pi}{7})
\big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n}  = {}\\
{} =  b_{2n}-b_{n}\, h_{2n-1}  + \tfrac{\sqrt{7}}{2}\, ( h_{2n-1}^{2} - h_{4n-1} )
:= \widetilde{B}_{n},
\end{multline}
%
%
\begin{multline}\label{lem2.6-w8}
2 \, \sin (\tfrac{2\, \pi}{7})\,
\big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n}
+ 2 \, \sin (\tfrac{4\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{2n} + {} \\
{}+ 2 \, \sin (\tfrac{8\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{2n}  = {}\\
{} = c_{2n}-c_{n}\, h_{2n-1}  + \tfrac{\sqrt{7}}{2}\, ( h_{2n-1}^{2} - h_{4n-1} )
:= \widetilde{C}_{n},
\end{multline}
\end{lemma}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Now we are ready to present the final result of this section.
All recurrent sequences defined in this section are applied below to the
description of the coefficients of certain polynomials.

\begin{theorem}\label{f2s-tw1}
The following decompositions of polynomials hold:
\begin{multline}\label{f2s-tw1-w-a}
\big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7})
\big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7})
\big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7})
\big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big) = {}\\
{} = \left\{
\begin{array}{ll}
\mathbb{X}^3 - a_k\,  \mathbb{X}^2 + 7\, B_{2k}\, \mathbb{X} + \sqrt{7}, & \quad \mbox{for } n=2\,k,\\
\mathbb{X}^3 - \alpha_{k-1}\,  \mathbb{X}^2 - 7\, B_{2k-1}\, \mathbb{X} + \sqrt{7}, & \quad \mbox{for } n=2\,k-1,\\
\end{array}
\right.
\end{multline}
%
%
\begin{multline}\label{f2s-tw1-w-b}
\big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7})
\big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7})
\big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7})
\big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big) = {}\\
{} = \left\{
\begin{array}{ll}
\mathbb{X}^3 - b_k\,  \mathbb{X}^2 + 7\,( C_{2k} - B_{2k} )\, \mathbb{X} + \sqrt{7}, & \quad \mbox{for } n=2\,k,\\
\mathbb{X}^3 - \beta_{k-1}\,  \mathbb{X}^2 + 7\, (B_{2k-1} - C_{2k-1})\, \mathbb{X} + \sqrt{7}, &
\quad \mbox{for } n=2\,k-1,\\
\end{array}
\right.
\end{multline}
%
%
\begin{multline}\label{f2s-tw1-w-c}
\big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7})
\big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7})
\big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7})
\big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} = \left\{
\begin{array}{ll}
\mathbb{X}^3 - c_k\,  \mathbb{X}^2 - 7\, C_{2k} \, \mathbb{X} + \sqrt{7}, & \quad \mbox{for } n=2\,k,\\
\mathbb{X}^3 - \gamma_{k-1}\,  \mathbb{X}^2 + 7\, C_{2k-1}\, \mathbb{X} + \sqrt{7}, &
\quad \mbox{for } n=2\,k-1,\\
\end{array}
\right.
\end{multline}
%
%
\begin{multline}\label{f2s-tw1-w-d}
\big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7})
\big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7})
\big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7})
\big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - f_n\,  \mathbb{X}^2 + (-1)^{n} \,
( 7\,{A}_{n} - 3\,\mathcal{A}_{n} ) \, \mathbb{X} - 1,
\end{multline}
%
%
\begin{multline}\label{f2s-tw1-w-e}
\big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7})
\big( 2\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7})
\big( 2\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7})
\big( 2\, \cos (\tfrac{2\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - g_n\,  \mathbb{X}^2 + (-1)^{n} \,
( \mathcal{A}_{n}  + \mathcal{A}_{n+1} - 7\, A_{n} ) \, \mathbb{X} - 1,
\end{multline}
%
%
\begin{multline}\label{f2s-tw1-w-f}
\big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big)\times {} \\
{} \times
\big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - {F}_n\,  \mathbb{X}^2 +
( f_{n}  + h_{n} ) \, \mathbb{X} - 1,
\end{multline}
%
%
\begin{multline}\label{f2s-tw1-w-g}
\big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big)\times {} \\
{} \times
\big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - {G}_n\,  \mathbb{X}^2 +
( f_{n}  + g_{n} ) \, \mathbb{X} - 1,
\end{multline}
%
%
\begin{multline}\label{f2s-tw1-w-h}
\big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7})
\big( 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7}) \big)^{n} \big) \times {} \\
{} \times
\big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - {H}_n\,  \mathbb{X}^2 +
( g_{n}  + h_{n} ) \, \mathbb{X} - 1.
\end{multline}
\end{theorem}

\begin{proof}
Ad~(\ref{f2s-tw1-w-d})
We have
\begin{equation*}
4 \,  \cos (\tfrac{4\, \pi}{7})
\, \cos (\tfrac{8\, \pi}{7})   =
2 \, \cos (\tfrac{2\, \pi}{7}) +  2 \, \cos (\tfrac{4\, \pi}{7})
\end{equation*}
hence, by~(\ref{w1-d}) we obtain
\begin{multline*}
\big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n+1} =
\big( 2 \, \cos (\tfrac{2\, \pi}{7}) +  2 \, \cos (\tfrac{4\, \pi}{7}) \big)
\big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n} = {} \\
{}=\Big[
\big(
2 \, \cos (\tfrac{2\, \pi}{7}) +  2 \, \cos (\tfrac{4\, \pi}{7})  + 2 \, \cos (\tfrac{8\, \pi}{7})
\big) -
\big(
 2 \, \cos (\tfrac{2\, \pi}{7}) +  2 \, \cos (\tfrac{8\, \pi}{7})
\big)
 + 2 \, \cos (\tfrac{2\, \pi}{7})
\Big]\times {} \\
{}\times
\big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n} %= {} \\
{}=
- \big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n}
- 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n} + {} \\
+
\big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n-1}.
\end{multline*}
Treating, in a similar way, the following products:
\begin{equation*}
\big( 4\, \cos (\tfrac{2\, \pi}{7}) \, \cos (\tfrac{4\, \pi}{7}) \big)^{n+1}
\qquad \mbox{ and } \qquad
\big( 4\, \cos (\tfrac{2\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n+1},
\end{equation*}
and adding all three received decompositions, by~(\ref{wzor-gw1new}),
we obtain
\begin{multline*}
(-1)^{n}\, \big( \mathcal{A}_{n+1} -  \mathcal{A}_{n} - \mathcal{A}_{n-1}\big) %= {}\\
 = 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \cos (\tfrac{4\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n} + {}\\
{} + 4\, \cos (\tfrac{2\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7}) \, \cos (\tfrac{4\, \pi}{7}) \big)^{n} %+ {}\\
 + 4\, \cos (\tfrac{4\, \pi}{7})\, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \cos (\tfrac{2\, \pi}{7}) \, \cos (\tfrac{8\, \pi}{7}) \big)^{n}.
\end{multline*}
But, by~\cite{wsw1}, we have
$$
\mathcal{A}_{n+3} - 2\, \mathcal{A}_{n+2} - \mathcal{A}_{n+1} + \mathcal{A}_{n} \equiv 0,
$$
hence
$$
\mathcal{A}_{n+1} - \mathcal{A}_{n} - \mathcal{A}_{n-1} = \mathcal{A}_{n} - \mathcal{A}_{n-2},
$$
which implies decomposition~(\ref{f2s-tw1-w-d}).
\end{proof}
\bigskip

\begin{remark}
{\rm
It should also be noted that the following interesting identity holds:
\begin{equation}
4\,  \mathcal{A}_{n} -  \mathcal{A}_{n-2} = 7\, A_{n}.
\end{equation}
}
\end{remark}



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\section{The second group of special cases of~(\ref{choinka})}

Let us set
\begin{align}
u_{n} &= 2\, \cos (\tfrac{2\, \pi}{7})
\big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} +
2\, \cos (\tfrac{4\, \pi}{7})
\big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} +
2\, \cos (\tfrac{8\, \pi}{7})
\big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.1}\\
v_{n} &= 2\, \cos (\tfrac{4\, \pi}{7})
\big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} +
2\, \cos (\tfrac{8\, \pi}{7})
\big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} +
2\, \cos (\tfrac{2\, \pi}{7})
\big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.2}\\
w_{n} &= 2\, \cos (\tfrac{8\, \pi}{7})
\big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} +
2\, \cos (\tfrac{2\, \pi}{7})
\big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} +
2\, \cos (\tfrac{4\, \pi}{7})
\big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.3}\\
x_{n} &= 2\, \sin (\tfrac{4\, \pi}{7})
\big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} +
2\, \sin (\tfrac{8\, \pi}{7})
\big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} +
2\, \sin (\tfrac{2\, \pi}{7})
\big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.4}\\
y_{n} &= 2\, \sin (\tfrac{8\, \pi}{7})
\big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} +
2\, \sin (\tfrac{2\, \pi}{7})
\big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} +
2\, \sin (\tfrac{4\, \pi}{7})
\big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.5}\\
z_{n} &=
\big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n+1} +
\big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n+1} +
\big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n+1},\label{w4.6}
\end{align}
for $n\in \mathbb{N}$ and $u_0=v_0=w_0=-1$
and $z_0=y_0=z_0=\sqrt{7}$, $z_1=7$ (see~Table~\ref{f2s-tab3},
and for sequence $\{-w_{2n+1}/\sqrt{7}\}$ see \seqnum{A115146}~\cite{sloan},
and see \seqnum{A079309}~\cite{sloan} for sequence $\{z_{2n}/\sqrt{7}\}$).

Then, as may by verified without difficulty,
the following recurrence relations hold
\begin{equation}
\left\{
\begin{array}{l}
u_{n+1} = x_{n},\\
v_{n+1} = -y_{n}-z_{n} = x_{n} - \sqrt{7} \, z_{n-1},\\
w_{n+1} = y_{n}-x_{n},\\
x_{n+1} = u_{n} - w_{n},\\
y_{n+1} = w_{n} - v_{n},\\
z_{n+1} = 2\, z_{n-1} - v_{n},
\end{array}
\right.
\end{equation}
for every $n\in \mathbb{N}$.
Hence,  we easily obtain
\begin{equation}
\left\{
\begin{array}{l}
x_{n+2} = x_{n} - w_{n+1} = 2\, x_{n} - y_{n},\\
y_{n+2} = y_{n} - v_{n+1} - x_{n} = 2\, y_{n}  - x_{n} + z_{n},\\
z_{n+2} = y_{n} + 3\, z_{n},
\end{array}
\right.
\end{equation}
for every $n\in \mathbb{N}$
and finally the recurrence relation (see also equation~(\ref{w1-b})):
\begin{equation}\label{w3n.9}
z_{n+6} - 7\, z_{n+4} + 14\, z_{n+2}  - 7\, z_{n} = 0,
\end{equation}
which also satisfies the remaining sequences discussed in this section:
$\{x_n\}$, $\{y_n\}$, $\{u_n\}$, $\{v_n\}$ and $\{w_n\}$.

\begin{remark}
{\rm
The characteristic polynomial of equation~(\ref{w3n.9}) (after rescaling)
has the form of~(\ref{w1-b})
and was recognized by Johannes Kepler (1571--1630). The roots of this polynomial are
equal to $|A_1A_2|^2$, $|A_1A_3|^2$ and
$|A_1A_4|^2$, where $A_1A_2\ldots A_7$ is
a~regular convex heptagon inscribed in the unit circle~\cite{GrzymkowskiWitula}.
}
\end{remark}


\begin{theorem}\label{f2s-tw2}
The following decompositions of polynomials hold:
\begin{multline}\label{f2s-tw2-w-a}
\big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7})
\big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7})
\big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7})
\big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - u_n\,  \mathbb{X}^2
+\tfrac{1}{2} \big( u_{n}^{2}  - v_{2n} - 2\, z_{2n-1} \big)\, \mathbb{X}
- (-\sqrt{7}\, )^{n},
\end{multline}
%
%
\begin{multline}\label{f2s-tw2-w-b}
\big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7})
\big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7})
\big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7})
\big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - v_n\,  \mathbb{X}^2
+\tfrac{1}{2} \big( v_{n}^{2}  - w_{2n} - 2\, z_{2n-1} \big)\, \mathbb{X}
- (-\sqrt{7}\, )^{n},
\end{multline}
%
%
\begin{multline}\label{f2s-tw2-w-c}
\big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7})
\big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7})
\big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7})
\big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - w_n\,  \mathbb{X}^2
+\tfrac{1}{2} \big( w_{n}^{2}  - u_{2n} - 2\, z_{2n-1} \big)\, \mathbb{X}
- (-\sqrt{7}\, )^{n},
\end{multline}
%
%
\begin{multline}\label{f2s-tw2-w-d}
\big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7})
\big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7})
\big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7})
\big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - x_n\,  \mathbb{X}^2
+\tfrac{1}{2} \big( x_{n}^{2}  + w_{2n} - 2\, z_{2n-1} \big)\, \mathbb{X}
- (-\sqrt{7}\, )^{n+1},
\end{multline}
%
%
\begin{multline}\label{f2s-tw2-w-e}
\big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7})
\big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7})
\big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7})
\big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - y_n\,  \mathbb{X}^2
+\tfrac{1}{2} \big( y_{n}^{2}  + u_{2n} - 2\, z_{2n-1} \big)\, \mathbb{X}
- (-\sqrt{7}\, )^{n+1},
\end{multline}
%
%
\begin{multline}\label{f2s-tw2-w-f}
\big( \mathbb{X} - \big( 2\, \sin (\tfrac{2\, \pi}{7}) \big)^{n+1} \big)
\big( \mathbb{X} - \big( 2\, \sin (\tfrac{4\, \pi}{7}) \big)^{n+1} \big)
\big( \mathbb{X} - \big( 2\, \sin (\tfrac{8\, \pi}{7}) \big)^{n+1} \big) = {}\\
{} =
\mathbb{X}^3 - z_n\,  \mathbb{X}^2
+\tfrac{1}{2} \big( z_{n}^{2} - z_{2n+1} \big)\, \mathbb{X}
- (-\sqrt{7}\, )^{n+1}.
\end{multline}
%
%
\end{theorem}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Additionally, primarily to describe the generating functions  of the sequences
$\{u_n\}$, $\{v_n\}$, $\{w_n\}$, $\{x_n\}$, $\{y_n\}$ and $\{z_n\}$
(see Section~\ref{roz-fun-two}) six new recurrence sequences were introduced, which,
from a certain point of view, are conjugated with sequences
$\{u_n\}$, \ldots, $\{z_n\}$.
Let us set
\begin{align}
u_{n}^{*} &= 2\, \cos (\tfrac{2\, \pi}{7})
\big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} +
2\, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} + {}\nonumber \\
&\hspace*{10mm} {}+
2\, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n},\label{w4.16a}\\
v_{n}^{*} &= 2\, \cos (\tfrac{2\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} +
2\, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} + {}\nonumber \\
&\hspace*{10mm} {}+
2\, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.17}\\
w_{n}^{*} &= 2\, \cos (\tfrac{2\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} +
2\, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} +{}\nonumber \\
&\hspace*{10mm} {}+
2\, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.18}\\
x_{n}^{*} &= 2\, \sin (\tfrac{2\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} +
2\, \sin (\tfrac{4\, \pi}{7})
\big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} +{}\nonumber \\
&\hspace*{10mm} {}+
2\, \sin (\tfrac{8\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.19}\\
y_{n}^{*} &= 2\, \sin (\tfrac{2\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} +
2\, \sin (\tfrac{4\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} +{}\nonumber \\
&\hspace*{10mm} {}+
2\, \sin (\tfrac{8\, \pi}{7})
\big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n},\label{w4.20}\\
z_{n}^{*} &= 2\, \sin (\tfrac{2\, \pi}{7})
\big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} +
2\, \sin (\tfrac{4\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} +{}\nonumber \\
&\hspace*{10mm} {}+
2\, \sin (\tfrac{8\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n},\label{w4.21}
\end{align}
for $n\in \mathbb{N}$ and $u_{0}^{*}=v_{0}^{*}=w_{0}^{*}=-1$
and $x_{0}^{*}=y_{0}^{*}=z_{0}^{*}=\sqrt{7}$ (see~Table~\ref{f2s-tab2}).


An easy computation shows that
the following recurrence relations hold
\begin{equation}
\left\{
\begin{array}{l}
u_{n+1}^{*} = u_{n}^{*}+w_{n}^{*}-v_{n}^{*}- z_{n-1}^{2} + z_{2n-1},\\
v_{n+1}^{*} = z_{n-1}^{2} - z_{2n-1} - u_{n}^{*},\\
w_{n+1}^{*} = u_{n}^{*}-w_{n}^{*},
\end{array}
\right.
\end{equation}
for every $n \geq 1$.
Hence,  we obtain
\begin{align}
&u_{n+1}^{*}+v_{n+1}^{*} = w_{n}^{*} - v_{n}^{*},\\
&u_{n}^{*} = w_{n+1}^{*} + w_{n}^{*},\\
&w_{n+2}^{*} + w_{n+1}^{*} - w_{n}^{*} = -v_{n+1}^{*} - v_{n}^{*},\\
&w_{n+3}^{*} -3\, w_{n+1}^{*} - w_{n}^{*} =
z_{2n+1} + z_{2n-1} -z_{n}^{2} - z_{n-1}^{2}.
\end{align}
for every $n \geq 1$.


Also the following recurrence relations hold
\begin{equation}
\left\{
\begin{array}{l}
x_{n+1}^{*} = 2\, y_{n}^{*} - z_{n}^{*},\\
y_{n+1}^{*} = 2\, x_{n}^{*}+y_{n}^{*}-z_{n}^{*},\\
z_{n+1}^{*} = -x_{n}^{*}-y_{n}^{*}-z_{n}^{*},
\end{array}
\right.
\end{equation}
for every $n \in \mathbb{N}$, which implies the identities
\begin{align}
&y_{n+1}^{*} = -x_{n+2}^{*} +2\, x_{n+1}^{*} + 7\, x_{n}^{*},\\
&x_{n+2}^{*} + 7\, x_{n}^{*} + 7\, x_{n-1}^{*} = 0
\end{align}
for every $n \geq 1$.


\begin{theorem}\label{f2s-tw2a}
The following decompositions of polynomials hold:
\begin{multline}\label{w4.30}
\big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7})
\big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) \times {}\\
{}\times \big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - u_{n}^{*}\,  \mathbb{X}^2
+(-\sqrt{7}\, )^{n}\,\big( u_{n}  + v_{n} \big)\, \mathbb{X}
- 7^{n},
\end{multline}
%
\begin{multline}\label{w4.31}
\big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) \times {}\\
{}\times
\big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - v_{n}^{*}\,  \mathbb{X}^2
+(-\sqrt{7}\, )^{n}\,\big( v_{n}  + w_{n} \big)\, \mathbb{X}
- 7^{n},
\end{multline}
%
\begin{multline}\label{w4.32}
\big( \mathbb{X} - 2 \, \cos (\tfrac{2\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \cos (\tfrac{4\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) \times {}\\
{}\times
\big( \mathbb{X} - 2 \, \cos (\tfrac{8\, \pi}{7})
\big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - w_{n}^{*}\,  \mathbb{X}^2
+(-\sqrt{7}\, )^{n}\,\big( u_{n}  + w_{n} \big)\, \mathbb{X}
- 7^{n},
\end{multline}
%
\begin{multline}\label{w4.33}
\big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7})
\big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) \times {}\\
{}\times
\big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - x_{n}^{*}\,  \mathbb{X}^2
+(-\sqrt{7}\, )^{n}\,\big( w_{n}  - v_{n} \big)\, \mathbb{X}
- 7^{n},
\end{multline}
%
\begin{multline}\label{w4.34}
\big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) \times {}\\
{}\times
\big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7})
\big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - y_{n}^{*}\,  \mathbb{X}^2
+(-\sqrt{7}\, )^{n}\,\big( u_{n}  - w_{n} \big)\, \mathbb{X}
- 7^{n},
\end{multline}
%
%
\begin{multline}\label{w4.35}
\big( \mathbb{X} - 2 \, \sin (\tfrac{2\, \pi}{7})
\big( 4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big)
\big( \mathbb{X} - 2 \, \sin (\tfrac{4\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{n} \big) \times {}\\
{}\times
\big( \mathbb{X} - 2 \, \sin (\tfrac{8\, \pi}{7})
\big( 4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{n} \big) = {}\\
{} =
\mathbb{X}^3 - z_{n}^{*}\,  \mathbb{X}^2
+(-\sqrt{7}\, )^{n}\,\big( v_{n}  - w_{n} \big)\, \mathbb{X}
- 7^{n}.
\end{multline}
%
\end{theorem}

\begin{lemma}
The following two groups of identities hold:
\begin{equation}
\begin{split}
u_{n}^{2} & = 2\, z_{2n-1} + v_{2n} + v_{n}^{*} + u_{n}^{*},\\
v_{n}^{2} & = 2\, z_{2n-1} + w_{2n} + v_{n}^{*} + w_{n}^{*},\\
w_{n}^{2} & = 2\, z_{2n-1} + u_{2n} + u_{n}^{*} + w_{n}^{*},\\
x_{n}^{2} & = 2\, z_{2n-1} - w_{2n} + w_{n}^{*} - v_{n}^{*},\\
y_{n}^{2} & = 2\, z_{2n-1} - u_{2n} + u_{n}^{*} - w_{n}^{*},\\
z_{n}^{2} & = 2\, z_{2n-1} + 2\, v_{n}^{'}  -2 \, u_{n}^{*},
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
\big(u_{n}^{*}\big)^{2} & = z_{2n-1}^{2} - z_{4n-1} + v_{n}^{*}
+ 2\, \big({-}\sqrt{7}\,\big)^{n} \big( u_{n} + v_{n} \big),\\
\big(v_{n}^{*}\big)^{2} & = z_{2n-1}^{2} - z_{4n-1} + w_{n}^{*}
+ 2\, \big({-}\sqrt{7}\,\big)^{n} \big( v_{n} + w_{n} \big),\\
\big(w_{n}^{*}\big)^{2} & = z_{2n-1}^{2} - z_{4n-1} + u_{n}^{*}
+ 2\, \big({-}\sqrt{7}\,\big)^{n} \big( u_{n} + w_{n} \big),\\
\big(x_{n}^{*}\big)^{2} & = z_{2n-1}^{2} - z_{4n-1} - w_{n}^{*}
+ 2\, \big({-}\sqrt{7}\,\big)^{n} \big( w_{n} - v_{n} \big),\\
\big(y_{n}^{*}\big)^{2} & = z_{2n-1}^{2} - z_{4n-1} - u_{n}^{*}
+ 2\, \big({-}\sqrt{7}\,\big)^{n} \big( u_{n} - w_{n} \big),\\
\big(z_{n}^{*}\big)^{2} & = z_{2n-1}^{2} - z_{4n-1} - v_{n}^{*}
+ 2\, \big({-}\sqrt{7}\,\big)^{n} \big( v_{n} - u_{n} \big).
\end{split}
\end{equation}
\end{lemma}

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\section{Some Ramanujan-type formulas}

Let $p,q,r\in \mathbb{R}$.
Shevelev~\cite{Shevelev}  (see also~\cite{GrzymkowskiWitula})
proved that if $z_1$, $z_2$, $z_3$ are roots of the polynomial
$$
w(z)=z^3+p\, z^2+q\, z+r
$$
and $z_1$, $z_2$, $z_3$ are all reals and at least the following condition holds
\begin{equation}\label{w-srf1}
p\, \sqrt[3]{r} + 3\, \big( \sqrt[3]{r} \big)^2 + q =0,
\end{equation}
then the following formula holds
\begin{multline}\label{w-srf2}
\sqrt[3]{z_1} + \sqrt[3]{z_2} + \sqrt[3]{z_3} =
\sqrt[3]{-p-6\,\sqrt[3]{r} + \sqrt[3]{(p+6\, \sqrt[3]{r})^3 - (p-3\, \sqrt[3]{r})^3} }= {}\\
{}=
\sqrt[3]{-p-6\,\sqrt[3]{r} + 3\, \sqrt[3]{9\, r + 3\, p\, (\sqrt[3]{r})^2 + p^2\, \sqrt[3]{r}} }
\end{multline}
(all radicals are determined to be real).


It was verified that only following polynomials (among all discussed in this paper) satisfy
the condition~(\ref{w-srf1}); simultanously below
the respective form of identity~(\ref{w-srf2})
is presented:
\bigskip

\noindent
polynomial (\ref{w1-d})  and polynomial~(\ref{wzor-gw1new}) for $n=1$
(it is the classical Ramanujan formula):
\begin{equation}\label{w-fib1}
\sqrt[3]{2\,\cos ( \tfrac{2\, \pi}{7})} +
\sqrt[3]{2\,\cos ( \tfrac{4\, \pi}{7})} +
\sqrt[3]{2\,\cos ( \tfrac{8\, \pi}{7})} =
\sqrt[3]{5-3\, \sqrt[3]{7}};
\end{equation}






\noindent
polynomial (\ref{f2s-tw1-w-e}) for $n=4$:
\begin{equation}\label{w-fib2}
\cos ( \tfrac{2\, \pi}{7})\, \sqrt[3]{\sec ( \tfrac{8\, \pi}{7})} +
\cos ( \tfrac{4\, \pi}{7})\, \sqrt[3]{\sec ( \tfrac{2\, \pi}{7})} +
\cos ( \tfrac{8\, \pi}{7})\, \sqrt[3]{\sec ( \tfrac{4\, \pi}{7})} =
\frac{1}{2}\, \sqrt[3]{18\, (2-\sqrt[3]{7})};
\end{equation}


\noindent
polynomial (\ref{f2s-tw1-w-e}) for $n=3$:
\begin{multline}\label{w-fib3}
2\,\cos ( \tfrac{2\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{4\, \pi}{7})} +
2\,\cos ( \tfrac{4\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{8\, \pi}{7})} +
2\,\cos ( \tfrac{8\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{2\, \pi}{7})} ={}\\
{}=
\sqrt[3]{-2-3\,\sqrt[3]{49}};
\end{multline}


\noindent
polynomial (\ref{f2s-tw1-w-f}) for $n=2,3$, respectively:
\begin{equation}\label{w-fib4}
\sqrt[3]{\frac{\cos ( \tfrac{2\, \pi}{7})}{1+\cos ( \tfrac{8\, \pi}{7})}} +
\sqrt[3]{\frac{\cos ( \tfrac{4\, \pi}{7})}{1+\cos ( \tfrac{2\, \pi}{7})}} +
\sqrt[3]{\frac{\cos ( \tfrac{8\, \pi}{7})}{1+\cos ( \tfrac{4\, \pi}{7})}}
=
\sqrt[3]{11-3\,\sqrt[3]{49}},
\end{equation}
\begin{equation}\label{w-fib5}
\sec ( \tfrac{4\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{2\, \pi}{7})} +
\sec ( \tfrac{8\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{4\, \pi}{7})} +
\sec ( \tfrac{2\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{8\, \pi}{7})} =
-2\,\sqrt[3]{9\,(1+\sqrt[3]{7})};
\end{equation}


\noindent
polynomial~(\ref{f2s-tw1-w-d}) for $n=1$,
polynomial (\ref{f2s-tw1-w-e}) for $n=1$ and polynomial (\ref{f2s-tw1-w-h}) for $n=2$:
\begin{equation}\label{w-fib6}
\sqrt[3]{\sec ( \tfrac{2\, \pi}{7})} +
\sqrt[3]{\sec ( \tfrac{4\, \pi}{7})} +
\sqrt[3]{\sec ( \tfrac{8\, \pi}{7})} =
\sqrt[3]{8-6\,\sqrt[3]{7}};
\end{equation}

\noindent
polynomial (\ref{f2s-tw2-w-b}) for $n=6$:
\begin{multline}\label{w-fib7}
\sin^2 ( \tfrac{2\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{4\, \pi}{7})} +
\sin^2 ( \tfrac{4\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{8\, \pi}{7})} +
\sin^2 ( \tfrac{8\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{2\, \pi}{7})}
={}\\
{}=
-\frac{1}{4}\,
\sqrt[3]{63\, (1+\sqrt[3]{7})};
\end{multline}


\noindent
polynomial (\ref{f2s-tw2-w-d}) for $n=2$:
\begin{equation}\label{w-fib8}
\sqrt[3]{\frac{\sin ( \tfrac{2\, \pi}{7})}{\sin ( \tfrac{8\, \pi}{7})}} +
\sqrt[3]{\frac{\sin ( \tfrac{4\, \pi}{7})}{\sin ( \tfrac{2\, \pi}{7})}} +
\sqrt[3]{\frac{\sin ( \tfrac{8\, \pi}{7})}{\sin ( \tfrac{4\, \pi}{7})}}
=
\sqrt[3]{5-3\,\sqrt[3]{7}};
\end{equation}


\noindent
polynomial (\ref{f2s-tw2-w-e}) for $n=2$:
\begin{equation}\label{w-fib9}
\sqrt[3]{\frac{\sin ( \tfrac{2\, \pi}{7})}{\sin ( \tfrac{4\, \pi}{7})}} +
\sqrt[3]{\frac{\sin ( \tfrac{4\, \pi}{7})}{\sin ( \tfrac{8\, \pi}{7})}} +
\sqrt[3]{\frac{\sin ( \tfrac{8\, \pi}{7})}{\sin ( \tfrac{2\, \pi}{7})}}
=
\sqrt[3]{4-3\,\sqrt[3]{7}};
\end{equation}


\noindent
polynomial (\ref{f2s-tw2-w-e}) for $n=5$:
\begin{equation}
\sin^2 (\tfrac{2\, \pi}{7})\, \sqrt[3]{\frac{\sin ( \tfrac{8\, \pi}{7})}{\sin ( \tfrac{2\, \pi}{7})}} +
\sin^2 (\tfrac{4\, \pi}{7})\, \sqrt[3]{\frac{\sin ( \tfrac{2\, \pi}{7})}{\sin ( \tfrac{4\, \pi}{7})}} +
\sin^2 (\tfrac{8\, \pi}{7})\, \sqrt[3]{\frac{\sin ( \tfrac{4\, \pi}{7})}{\sin ( \tfrac{8\, \pi}{7})}}
=
\frac{1}{4}\,
\sqrt[3]{77-21\,\sqrt[3]{49}};
\end{equation}



\noindent
polynomial (\ref{w4.30}) for $n=3$:
\begin{multline}
\sqrt[6]{7}\, \Big(
\csc(\tfrac{2\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{2\, \pi}{7})} +
\csc(\tfrac{4\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{4\, \pi}{7})} +
\csc(\tfrac{8\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{8\, \pi}{7})}
\Big)
={}\\
{}=
2\,
\sqrt[3]{2+3\,\sqrt[3]{49}};
\end{multline}



\noindent
polynomial (\ref{w4.32}) for $n=3$:
\begin{multline}
\sqrt[6]{7}\, \Big(
\csc(\tfrac{4\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{2\, \pi}{7})} +
\csc(\tfrac{8\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{4\, \pi}{7})} +
\csc(\tfrac{2\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{8\, \pi}{7})}
\Big)
={}\\
{}=
2\,
\sqrt[3]{3\,\sqrt[3]{7}-5};
\end{multline}



\noindent
polynomial (\ref{w4.32}) for $n=6$:
\begin{multline}
\csc^2(\tfrac{4\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{2\, \pi}{7})} +
\csc^2(\tfrac{8\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{4\, \pi}{7})} +
\csc^2(\tfrac{2\, \pi}{7})\, \sqrt[3]{2\cos ( \tfrac{8\, \pi}{7})}
={}\\
{}=
-4\,
\sqrt[3]{\frac{2+3\,\sqrt[3]{49}}{7}}.
\end{multline}



\begin{remark}
{\rm
Shevelev~\cite{Shevelev}
presented
the identities~(\ref{w-fib8}) and~(\ref{w-fib9})
in the following alternative form:
\begin{equation}
\sqrt[3]{2+\sec ( \tfrac{2\, \pi}{7})} +
\sqrt[3]{2+\sec ( \tfrac{4\, \pi}{7})} +
\sqrt[3]{2+\sec ( \tfrac{8\, \pi}{7})} =
\sqrt[3]{6\,\sqrt[3]{7}-10}
\end{equation}
and
\begin{multline}
\sqrt[3]{1+2\,\cos ( \tfrac{2\, \pi}{7})} +
\sqrt[3]{1+2\,\cos ( \tfrac{4\, \pi}{7})} +
\sqrt[3]{1+2\,\cos ( \tfrac{8\, \pi}{7})}
={}\\
{}=
\sqrt[3]{\frac{2\cos ( \tfrac{2\, \pi}{7})}{2\cos ( \tfrac{2\, \pi}{7})+1}} +
\sqrt[3]{\frac{2\cos ( \tfrac{4\, \pi}{7})}{2\cos ( \tfrac{4\, \pi}{7})+1}} +
\sqrt[3]{\frac{2\cos ( \tfrac{8\, \pi}{7})}{2\cos ( \tfrac{8\, \pi}{7})+1}}
=
\sqrt[3]{3\,\sqrt[3]{7}-4},
\end{multline}
respectively.
}
\end{remark}



\begin{remark}
{\rm
By~(\ref{w-fib7}),~(\ref{w-fib1})  and by applied the formula
$2\, \sin^2 \alpha = 1-\cos (2\alpha)$
we obtain
\begin{multline}
2\, \cos \big( \tfrac{2\, \pi}{7} \big)\,
\sqrt[3]{2\, \cos \big( \tfrac{2\, \pi}{7} \big)\,} +
2\, \cos \big( \tfrac{4\, \pi}{7} \big)\,
\sqrt[3]{2\, \cos \big( \tfrac{4\, \pi}{7} \big)\,} +
2\, \cos \big( \tfrac{8\, \pi}{7} \big)\,
\sqrt[3]{2\, \cos \big( \tfrac{8\, \pi}{7} \big)\,} = {} \\
{}=
\sqrt[3]{63\, \big(1+\sqrt[3]{7}\big)\,}
-2\,\sqrt[3]{5 - 3\, \sqrt[3]{7}\,}.
\end{multline}


Moreover, from~(\ref{w-fib1}),~(\ref{w-fib6})  and the equality
$8\, \cos (\frac{2\pi}{7}) \, \cos (\frac{4\pi}{7}) \, \cos (\frac{8\pi}{7}) = 1$ we get
\begin{multline}
\sqrt[3]{ \Big(2\, \cos \big( \tfrac{2\, \pi}{7} \big)\Big)^2\,} +
\sqrt[3]{\Big(2\, \cos \big( \tfrac{4\, \pi}{7} \big)\Big)^2\,} +
\sqrt[3]{\Big(2\, \cos \big( \tfrac{8\, \pi}{7} \big)\Big)^2\,} = {} \\
{}=
\sqrt[3]{\Big( 5 - 3\, \sqrt[3]{7} \Big)^2\,}
-2\,\sqrt[3]{4 - 3\, \sqrt[3]{7}\,}.
\end{multline}
}
\end{remark}

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\section{The sine-Fibonacci numbers of order $7$}

The quasi-Fibonacci numbers of order $7$ introduced
by Witu{\l}a et al.\ in~\cite{wsw1} and further developed in Section~\ref{roz1}
constitute the simplest, only one-parameter type of the so-called
cosine-Fibonacci numbers of order $7$.
The name ``cosine-Fibonacci numbers'' is derived from the
form of the decomposition of formulas:
$(1+\delta\, (\xi+\xi^6))^{n}=(1+2\, \delta\, \cos(\tfrac{2\pi}{7}))^{n}$,~\ldots
In this Section we shall introduce and analyze a sine variety of these numbers,
created in the course of decomposing formulas:
$(1+\delta\, (\xi-\xi^6))^{n}=(1+2\, i\, \delta\, \sin(\tfrac{2\pi}{7}))^{n}$,~\ldots


\begin{theorem}
Let $\xi, \delta \in \mathbb{C}$, $\xi^7=1$ and $\xi \neq 1$. Then,
for every $n\in \mathbb{N}$, there exist polynomials
$p_n, r_n, s_n, k_n, l_n \in \mathbb{Z}[\delta]$,
called here the sine-Fibonacci numbers of order $7$,
so that
\begin{align}
(1+\delta\, (\xi^{\phantom{2}}-\xi^6))^n &=
p_n(\delta)+r_n(\delta)\, (\xi^{\phantom{2}}-\xi^6)+s_n(\delta)\, (\xi^2-\xi^5)+{}\nonumber\\
&\phantom{=}\hspace*{2ex}{}
+k_n(\delta)\, (\xi^{\phantom{2}}+\xi^6)+l_n(\delta)\, (\xi^2+\xi^5),\\
(1+\delta\,  (\xi^2 -\xi^5))^n&=
p_n(\delta)+r_n(\delta)\, (\xi^{2}-\xi^5)+s_n(\delta)\, (\xi^4-\xi^3)+{}\nonumber\\
&\phantom{=}\hspace*{2ex}{}
+k_n(\delta)\, (\xi^{2}+\xi^5)+l_n(\delta)\, (\xi^4+\xi^3),\\
(1+\delta\,  (\xi^4 -\xi^3))^n&=
p_n(\delta)+r_n(\delta)\, (\xi^{4}-\xi^3)+s_n(\delta)\, (\xi^{\phantom{2}}-\xi^6)+{}\nonumber\\
&\phantom{=}\hspace*{2ex}{}
+k_n(\delta)\, (\xi^{4}+\xi^3)+l_n(\delta)\, (\xi^{\phantom{2}}+\xi^6).
\end{align}
We have $p_1(\delta)=1$, $r_1(\delta)=\delta$ and
$s_1(\delta)=k_1(\delta)=l_1(\delta)=0$ (see Table~\ref{f2s-tab4}, where the initial
values of these polynomial are presented).
These polynomials are connected by recurrence relations
\begin{equation}\label{f2s-r6-w1g}
\left\{
\begin{array}{l}
p_{n+1}(\delta) = p_{n}(\delta) - \delta\, s_{n}(\delta) -
2\, \delta\, r_{n}(\delta) - i\, \sqrt{7}\, \delta l_{n}(\delta),\\
r_{n+1}(\delta) = r_{n}(\delta) + \delta\, p_{n}(\delta),\\
s_{n+1}(\delta) = s_{n}(\delta) + \delta\, k_{n}(\delta) + \delta\, l_{n}(\delta),\\
k_{n+1}(\delta) = k_{n}(\delta) - 2\, \delta\, s_{n}(\delta),\\
l_{n+1}(\delta) = l_{n}(\delta) + \delta\, r_{n}(\delta) - \delta\, s_{n}(\delta),\\
\end{array}
\right.
\end{equation}
for $n\in \mathbb{N}$. Hence, the following relationships can be deduced:
\begin{align}
\delta\, p_{n}(\delta) & = r_{n+1}(\delta) - r_{n}(\delta),\label{w5.4.a}\\
2\,\delta\, s_{n}(\delta) & = k_{n+1}(\delta) - k_{n}(\delta),\label{w5.4.b}\\
2\,\delta^2\, l_{n}(\delta) & = -k_{n+2}(\delta) +2 \, k_{n+1}(\delta)
-(1+2\, \delta^2)\, k_{n}(\delta),\label{w5.4.c}\\
2\,\delta^3\, r_{n}(\delta) & = -k_{n+3}(\delta) + 3 \, k_{n+2}(\delta)
-3\,(1+\delta^2)\, k_{n+1}(\delta)
+(1+3\, \delta^2)\, k_{n}(\delta)\label{w5.4.d}
\end{align}
and, at last, the main identity
\begin{multline}\label{f2s-r6-wdelta}
k_{n+5}(\delta) - 5\, k_{n+4} + (10+5\, \delta^2)\, k_{n+3}(\delta)+
(i\, \sqrt{7} \, \delta^3  -15\, \delta^2 -10)\, k_{n+2}(\delta)+\\
+(7\, \delta^4 - i\, 2\, \sqrt{7} \, \delta^3  + 15\, \delta^2 + 5)\, k_{n+1}(\delta)+\\
+(i\, 2\, \sqrt{7} \, \delta^5 - 7\, \delta^4 + i\, \sqrt{7} \, \delta^3  -
5\, \delta^2 - 1)\, k_{n}(\delta)
=0.
\end{multline}
This identity satisfies, also by~(\ref{w5.4.a})--(\ref{w5.4.d}), the remaining
polynomials: $p_{n}(\delta)$, $r_{n}(\delta)$, $s_{n}(\delta)$, and $l_{n}(\delta)$,
$n\in \mathbb{N}$. The characteristic polynomial  corresponding to identity~(\ref{f2s-r6-wdelta})
has the following decomposition:
\begin{multline}\label{wz6.6}
\mathbb{X}^5- 5\, \mathbb{X}^4 + (10+5\, \delta^2)\, \mathbb{X}^3+
(i\, \sqrt{7} \, \delta^3  -15\, \delta^2 -10)\, \mathbb{X}^2
+(7\, \delta^4 - i\, 2\, \sqrt{7} \, \delta^3  + 15\, \delta^2 + 5)\, \mathbb{X} + {}\\
{}+
i\, 2\, \sqrt{7} \, \delta^5 - 7\, \delta^4 + i\, \sqrt{7} \, \delta^3  - 5\, \delta^2 - 1= %{}\\
%{} =
(\mathbb{X}-1-\delta\, (\xi-\xi^6))
(\mathbb{X}-1-\delta\, (\xi^2-\xi^5)) \times{}\\
{}\times
(\mathbb{X}-1-\delta\, (\xi^4-\xi^3))
(\mathbb{X}-1+\delta\, (\xi+\xi^2+\xi^4))
(\mathbb{X}-1+\delta\, (\xi^3+\xi^5+\xi^6)) = {} \\
{} =
\big( \mathbb{X}^3 + (-3 - i\, \sqrt{7}\, \delta) \, \mathbb{X}^2 +
(3 + i\, 2\, \sqrt{7}\, \delta)\, \mathbb{X}
-i\, \sqrt{7}\, \delta^3 - i\, \sqrt{7}\, \delta -1 \big) \times {}\\
{}\times
\big( \mathbb{X}^2 + (-2 + i\, \sqrt{7}\, \delta) \, \mathbb{X} +
1 - 2\, \delta^2 - i\, \sqrt{7}\, \delta  \big).
\end{multline}
Hence, we obtain, for example, the following explicit form of $k_{n}(\delta)$:
\begin{multline}
k_{n}(\delta) =
a\,(1+\delta\, (\xi-\xi^6))^{n} +
b\,(1+\delta\, (\xi^2-\xi^5))^{n} + {}\\
{} +
c\,(1+\delta\, (\xi^4-\xi^3))^{n} +
d\,(1-\delta\, (\xi+\xi^2+\xi^4))^{n} +
e\,(1+\delta\, (\xi^3+\xi^5+\xi^6))^{n},
\end{multline}
where
\begin{align*}
a & = -2 \big( 5+(\xi^2+\xi^5) +7\, (\xi^3+\xi^4) \big)^{-1}
=\tfrac{2}{301} \big( 46\, (\xi+\xi^6)+5\,(\xi^3+\xi^4) -11\big),\\
b & = \phantom{-}2 \big( 2+7\, (\xi^2+\xi^5) +6\, (\xi^3+\xi^4) \big)^{-1}
=\tfrac{2}{187} \big(19\, (\xi+\xi^6)+46\,(\xi^2+\xi^5) -5\big),\\
c & = -2 \big( 4+6\, (\xi^2+\xi^5) - (\xi^3+\xi^4) \big)^{-1}
=\tfrac{2}{301} \big(5\, (\xi^2+\xi^5)+46\,(\xi^3+\xi^4) -11\big),\\
d & = -2 \big( {-5}+2\, (\xi+\xi^2+\xi^4) \big)^{-1}=\tfrac{2}{43} \big( 5-2\, (\xi^3+\xi^5+\xi^6) \big) ,\\
e & = \phantom{-}2 \big( 7+2\, (\xi+\xi^2+\xi^4) \big)^{-1}=\tfrac{2}{43} \big( 7+2\, (\xi^3+\xi^5+\xi^6) \big).
\end{align*}
\end{theorem}


\begin{corollary}
The following identity holds:
\begin{multline}\label{wz6.8}
\Omega_{n} (\delta) :=
(1+\delta\, (\xi-\xi^6))^{n} +
(1+\delta\, (\xi^2-\xi^5))^{n} +
(1+\delta\, (\xi^4-\xi^3))^{n} = {}\\
{} =
3\, p_{n}(\delta) - k_{n}(\delta) - l_{n}(\delta)
+ i\, \sqrt{7} \big( r_{n}(\delta) + s_{n}(\delta) \big).
\end{multline}
\end{corollary}

\begin{definition}
To simplify the notation, we shall write $\Omega_n$ instead of $\Omega_{n}(1)$.
\end{definition}


\begin{theorem}
The following decompositions of the polynomials hold:
\begin{multline}
q_{n}(\mathbb{X}; \delta):=\\
=\big(\mathbb{X}-\big(1+\delta\, (2\, i\, \sin (\tfrac{2\, \pi}{7}))^{n}\big)\big)
\big(\mathbb{X}-\big(1+\delta\, (2\, i\, \sin (\tfrac{4\, \pi}{7}))^{n}\big)\big)
\big(\mathbb{X}-\big(1+\delta\, (2\, i\, \sin (\tfrac{8\, \pi}{7}))^{n}\big)\big)
 = {}\\
{} =
\mathbb{X}^3 -  \Omega_{n} (\delta)\, \mathbb{X}^2
+\tfrac{1}{2} \big( ( \Omega_{n} (\delta) )^{2} - \Omega_{2n} (\delta) \big)\, \mathbb{X}
- \big(1+i\, \sqrt{7} \, \delta + i\, \sqrt{7} \, \delta^3 \big)^{n},\label{w5n.9}
\end{multline}
%
\begin{multline}\label{w6.14}
\big(x- \big( \cot (\tfrac{2\, \pi}{7}) \big)^{n}\big)
\big(x- \big( \cot (\tfrac{4\, \pi}{7}) \big)^{n}\big)
\big(x- \big( \cot (\tfrac{8\, \pi}{7}) \big)^{n}\big)
 = {}\\
{} = x^3 - \big( \tfrac{3}{\sqrt{7}} \big)^{n} \,
\mathcal{A}_{n}(\tfrac{2}{3})\, x^2 +  \Omega_{n}
(\tfrac{2\, i}{\sqrt{7}})\, x +\frac{(-1)^{n-1}}{\big(\sqrt{7}\,
\big)^{n}}.
\end{multline}
%
Moreover, the following identity holds:
\begin{multline}
\Omega_{n}^{2}(\delta) = \Omega_{2n}(\delta) +
2\, \big( 3\, p_{n}^{2}(\delta) - 2\, l_{n}^{2}(\delta) - 2\, k_{n}^{2}(\delta)
-2\, p_{n}(\delta)\, k_{n}(\delta) - 2\, p_{n}(\delta)\, l_{n}(\delta)
- 7 \, r_{n}(\delta)\, s_{n}(\delta) + {} \\
{} + 3\, k_{n}(\delta)\, l_{n}(\delta)
+ i\, 2\, \sqrt{7}\, ( p_{n}(\delta)\, r_{n}(\delta) + p_{n}(\delta)\, s_{n}(\delta)
- r_{n}(\delta)\, k_{n}(\delta) - s_{n}(\delta)\, l_{n}(\delta) ) + {} \\
{} + i\, \sqrt{7} ( r_{n}(\delta)\, l_{n}(\delta) - k_{n}(\delta)\, s_{n}(\delta) ) \big)
= {} \\
{} =
\Omega_{2n}(\delta) + 4\, p_{n}(\delta)\, \Omega_{n}(\delta) +
2\, \big( -3\, p_{n}^{2}(\delta) - 2\, l_{n}^{2} (\delta)- 2\, k_{n}^{2}(\delta)
- 7\, r_{n}(\delta)\, s_{n}(\delta) + 3\, k_{n}(\delta)\, l_{n}(\delta) + {} \\
{} + i\, \sqrt{7} ( r_{n}(\delta)\, l_{n}(\delta) - 2\, r_{n}(\delta)\, k_{n}(\delta)
- k_{n}(\delta)\, s_{n}(\delta)  - 2\, s_{n}(\delta)\, l_{n}(\delta)) \big).
\end{multline}
\end{theorem}

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%%%%%%%%%%%%%%   Section 7   %%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Generating functions}\label{roz-fun-two}

The generating functions of almost all sequences discussed
and defined in this paper are presented in this section.




By~(\ref{w-and}) and~(\ref{w1.10}) we obtain (for~$k\in \mathbb{N}$):
\begin{multline}
\sum_{n=0}^{\infty}
\mathcal{A}_{k n}(\delta)\, \mathbb{X}^{n} =
\big( 1-\big(1+\delta\,(\xi+\xi^6)\big)^{k}\, \mathbb{X} \big)^{-1} +
\big( 1-\big(1+\delta\,(\xi^2+\xi^5)\big)^{k}\, \mathbb{X} \big)^{-1} + {} \\
{} + \big( 1-\big(1+\delta\,(\xi^3+\xi^4)\big)^{k}\, \mathbb{X} \big)^{-1} =
\frac{3 - 2\, \mathcal{A}_{k}(\delta) \, \mathbb{X} +
\mathcal{B}_{k}(\delta)\, \mathbb{X}^2}{1-
\mathcal{A}_{k}(\delta)\, \mathbb{X} +  \mathcal{B}_{k}(\delta)\, \mathbb{X}^2
-  (1-\delta-2\, \delta^2+\delta^3)^{k}\, \mathbb{X}^3},
\end{multline}
and, in the sequel, for $k=1$ we get
\begin{multline}
\sum_{n=0}^{\infty}
\mathcal{A}_{n}(\delta)\, \mathbb{X}^{n} =
\big( 1-\big(1+\delta\,(\xi+\xi^6)\big)\, \mathbb{X} \big)^{-1} +
\big( 1-\big(1+\delta\,(\xi^2+\xi^5)\big)\, \mathbb{X} \big)^{-1} + {} \\
{} + \big( 1-\big(1+\delta\,(\xi^3+\xi^4)\big)\, \mathbb{X} \big)^{-1}
=\frac{\mathbb{X}^2\, p_{7}^{'}\big( 1/\mathbb{X}; \delta\big)}{\mathbb{X}^3\, p_{7}\big( 1/\mathbb{X}; \delta\big)} = {} \\
{} =
\frac{3+2\, (\delta-3)\, \mathbb{X} +  (3-2\, \delta-2\, \delta^2)\, \mathbb{X}^2}{1+
(\delta-3)\, \mathbb{X} +  (3-2\, \delta-2\, \delta^2)\, \mathbb{X}^2
+  (-1+\delta+2\, \delta^2-\delta^3)\, \mathbb{X}^3},
\end{multline}
where $p_{7}(\mathbb{X},\delta)=\mathbb{X}^3+(\delta -3)\mathbb{X}^2+
(3-2\delta -2\delta^2)\mathbb{X} +
(-1+\delta +2\delta^2-\delta^3)$ (see~\cite{wsw1}).

By~(\ref{w-bnd}) and~(\ref{w1.19}) we get (for~$k\in \mathbb{N}$):
\begin{multline}
\sum_{n=0}^{\infty}
\mathcal{B}_{k n}(\delta)\, \mathbb{X}^{n} =
\frac{\mathbb{X}^2\, r_{k}^{'}\big( 1/\mathbb{X}; \delta\big)}{\mathbb{X}^3\, r_{k}\big( 1/\mathbb{X}; \delta\big)}
 = {} \\
{} =
\frac{3-2\, \mathcal{B}_{k}(\delta)\, \mathbb{X} +
(1-\delta-2\, \delta^2+\delta^3)\,\mathcal{A}_{k}(\delta)\, \mathbb{X}^2}{1-
\mathcal{B}_{k}(\delta)\, \mathbb{X} +  (1-\delta-2\, \delta^2+\delta^3)\,\mathcal{A}_{k}(\delta)\, \mathbb{X}^2
-  (1-\delta-2\, \delta^2+\delta^3)^{2 k}\, \mathbb{X}^3},
\end{multline}
where $r_{n}(\mathbb{X},\delta)$ is defined in~(\ref{w1.19}).
By~(3.19) from~\cite{wsw1} we obtain
\begin{multline}
\sum_{n=0}^{\infty}
{A}_{n}(\delta)\, \mathbb{X}^{n} =
\frac{2-\xi^3-\xi^4}{1-\big(1+\delta\,(\xi+\xi^6)\big)\, \mathbb{X}} +
\frac{2-\xi-\xi^6}{1-\big(1+\delta\,(\xi^2+\xi^5)\big)\, \mathbb{X}} + {}\\
{} + \frac{2-\xi^2-\xi^5}{1-\big(1+\delta\,(\xi^3+\xi^4)\big)\, \mathbb{X}} =
\frac{1+(-2+\delta)\, \mathbb{X} +
(1-\delta)\, \mathbb{X}^2}{\mathbb{X}^3\, p_{7}\big( 1/\mathbb{X}; \delta\big)};
\end{multline}
by~(3.18) and~(3.17) from~\cite{wsw1} we obtain respectively:
\begin{multline}
\sum_{n=0}^{\infty}
{B}_{n}(\delta)\, \mathbb{X}^{n} =
\frac{\xi+\xi^6-\xi^3-\xi^4}{1-\big(1+\delta\,(\xi+\xi^6)\big)\, \mathbb{X}} +
\frac{\xi^2+\xi^5-\xi-\xi^6}{1-\big(1+\delta\,(\xi^2+\xi^5)\big)\, \mathbb{X}} + {} \\
{} + \frac{\xi^3+\xi^4-\xi^2-\xi^5}{1-\big(1+\delta\,(\xi^3+\xi^4)\big)\, \mathbb{X}}
=
\frac{\delta\, \mathbb{X} + (\delta^2-\delta)\, \mathbb{X}^2}{\mathbb{X}^3\, p_{7}\big( 1/\mathbb{X}; \delta\big)}
\end{multline}
and
\begin{equation}
\sum_{n=0}^{\infty}
{C}_{n}(\delta)\, \mathbb{X}^{n} =
\sum_{n=0}^{\infty} \big(3 A_{n}(\delta) - B_{n}(\delta) - \mathcal{A}_{n}(\delta) \big)\, \mathbb{X}^{n} =
\frac{\delta^2\, \mathbb{X}^2}{\mathbb{X}^3\, p_{7}\big( 1/\mathbb{X}; \delta\big)}.
\end{equation}
%
Equally, on the grounds of (\ref{w2.1})--(\ref{w2.3})
and (\ref{lem2.6-w6})--(\ref{lem2.6-w8}), we obtain the following formulas:
\begin{equation}
\sum_{n=0}^{\infty}
a_{k n}\, \mathbb{X}^{n} =
\frac{\sqrt{7}- (b_k + c_k )\,\mathbb{X}+ \widetilde{A}_{k} \,\mathbb{X}^2}{1-
\mathcal{B}_{2k}\, \mathbb{X} + \mathcal{A}_{2k}\, \mathbb{X}^2-\mathbb{X}^3},
\end{equation}

\begin{equation}
\sum_{n=0}^{\infty}
b_{k n}\, \mathbb{X}^{n} =
\frac{\sqrt{7}- (a_k + c_k )\,\mathbb{X}+ \widetilde{B}_{k} \,\mathbb{X}^2}{1-
\mathcal{B}_{2k}\, \mathbb{X} + \mathcal{A}_{2k}\, \mathbb{X}^2-\mathbb{X}^3},
\end{equation}

\begin{equation}
\sum_{n=0}^{\infty}
c_{k n}\, \mathbb{X}^{n} =
\frac{\sqrt{7}- (a_k + b_k )\,\mathbb{X}+ \widetilde{C}_{k} \,\mathbb{X}^2}{1-
\mathcal{B}_{2k}\, \mathbb{X} + \mathcal{A}_{2k}\, \mathbb{X}^2-\mathbb{X}^3}.
\end{equation}

The special cases (for $k=1$) can also be generated in the following selective ways:

\noindent
by~(\ref{w2.1}) and~(\ref{w1-e}) we obtain
\begin{multline}
\sum_{n=0}^{\infty}
a_{n}\, \mathbb{X}^{n} =
\frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-4\, \cos^2 \big(\frac{8\, \pi}{7}\big)\, \mathbb{X} } +
\frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-4\, \cos^2 \big(\frac{2\, \pi}{7}\big)\, \mathbb{X} } +
\frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-4\, \cos^2 \big(\frac{4\, \pi}{7}\big)\, \mathbb{X} }  = {}\\
{}=
\frac{\sqrt{7}-2\, \sqrt{7}\,\mathbb{X}- \sqrt{7}\,\mathbb{X}^2}{1-5\, \mathbb{X} +
6\, \mathbb{X}^2-\mathbb{X}^3};
\end{multline}
however, by~(\ref{w2.2}) and~(\ref{f2s-w1g}) we obtain
\begin{multline}
\sum_{n=0}^{\infty}
b_{n}\, \mathbb{X}^{n} =
\sum_{n=0}^{\infty}
\big( a_{n+1} +2\, a_n \big) \, \mathbb{X}^{n} =
\frac{1+2\, \mathbb{X}}{\mathbb{X}}
\Big( \sum_{n=0}^{\infty} a_{n}\, \mathbb{X}^{n} \Big)
-\frac{\sqrt{7}}{\mathbb{X}}={}\\
{}=
\frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-4\, \cos^2 \big(\frac{8\, \pi}{7}\big)\, \mathbb{X} } +
\frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-4\, \cos^2 \big(\frac{2\, \pi}{7}\big)\, \mathbb{X} } +
\frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-4\, \cos^2 \big(\frac{4\, \pi}{7}\big)\, \mathbb{X} }  = {}\\
{}=
\frac{\sqrt{7}-3\, \sqrt{7}\,\mathbb{X}+3\, \sqrt{7}\,\mathbb{X}^2}{1-5\, \mathbb{X} + 6\, \mathbb{X}^2-\mathbb{X}^3}
\end{multline}
and
\begin{equation}
\sum_{n=0}^{\infty}
c_{n}\, \mathbb{X}^{n} =
\sum_{n=0}^{\infty}
\big( a_n +2\, b_n - b_{n+1}\big) \, \mathbb{X}^{n} =
\frac{\sqrt{7}-5\, \sqrt{7}\,\mathbb{X}+4\, \sqrt{7}\,\mathbb{X}^2}{1-5\, \mathbb{X} +
6\, \mathbb{X}^2-\mathbb{X}^3}.
\end{equation}




By~(\ref{wz6.8}) and~(\ref{w5n.9}) we get
\begin{multline}
\sum_{n=0}^{\infty}
\Omega_{k n}(\delta)\, \mathbb{X}^{n} =
\big( 1-\big(1+\delta\,(\xi-\xi^6)\big)^k\, \mathbb{X} \big)^{-1} +
\big( 1-\big(1+\delta\,(\xi^2-\xi^5)\big)^k\, \mathbb{X} \big)^{-1} + {}\\
{}+\big( 1-\big(1+\delta\,(\xi^3-\xi^4)\big)^k\, \mathbb{X} \big)^{-1} =
\frac{\mathbb{X}^2\, q_{k}^{'}\big( 1/\mathbb{X}; \delta\big)}{\mathbb{X}^3\, q_{k}\big( 1/\mathbb{X}; \delta\big)}
= {} \\
{} = \frac{3-2\, \Omega_{k}(\delta)\, \mathbb{X} +
\tfrac{1}{2}\,\big( (\Omega_{k}(\delta))^{2} - \Omega_{2k}(\delta) \big)\, \mathbb{X}^2}{1-
\Omega_{k}(\delta)\, \mathbb{X} +
\tfrac{1}{2}\,\big( (\Omega_{k}(\delta))^{2} - \Omega_{2k}(\delta) \big)\, \mathbb{X}^2
-  (1+i\, \sqrt{7}\,\delta+i\,\sqrt{7}\, \delta^3)^{k}\, \mathbb{X}^3},
\end{multline}
and, the special case, for $k=1$
(by~(\ref{wz6.8}) and~(\ref{wz6.6})):
\begin{multline}
\sum_{n=0}^{\infty}
\Omega_{n}(\delta)\, \mathbb{X}^{n} =
\big( 1-\big(1+\delta\,(\xi-\xi^6)\big)\, \mathbb{X} \big)^{-1} +
\big( 1-\big(1+\delta\,(\xi^2-\xi^5)\big)\, \mathbb{X} \big)^{-1} + {} \\
{} + \big( 1-\big(1+\delta\,(\xi^4-\xi^3)\big)\, \mathbb{X} \big)^{-1}
=\frac{\mathbb{X}^2\, q_{1}^{'}\big( 1/\mathbb{X}; \delta\big)}{\mathbb{X}^3\, q_{1}\big( 1/\mathbb{X}; \delta\big)} = {} \\
{} =
\frac{3-2\, (3+i\, \sqrt{7}\,\delta)\, \mathbb{X} +  (3+i\,2\,\sqrt{7}\,\delta)\, \mathbb{X}^2}{1-
(3+i\, \sqrt{7}\,\delta)\, \mathbb{X} +  (3+i\,2\,\sqrt{7}\,\delta)\, \mathbb{X}^2
-  (1+i\, \sqrt{7}\,\delta+i\,\sqrt{7}\, \delta^3)\, \mathbb{X}^3},
\end{multline}
where $q_{k}(\mathbb{X};\delta)$ is defined in~(\ref{w5n.9}).

By~(\ref{w1-d}), (\ref{w2.11a})--(\ref{w2.11}) and~(\ref{lem2.6-w1})--(\ref{lem2.6-w3})
the following formulas may be obtained:
\begin{multline}\label{w7-15}
\sum_{n=0}^{\infty}
f_{k n}\, \mathbb{X}^{n} =
\frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \cos (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \cos (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \cos (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} }  = {}\\
{} =
\frac{-1-( g_{k} + h_{k} )\,\mathbb{X}+ {F}_{k}\,\mathbb{X}^2}{1-
h_{k-1}\, \mathbb{X} + \tfrac{1}{2}  \big( h_{k-1}^{2} - h_{2k-1} \big)\, \mathbb{X}^2
-\mathbb{X}^3},
\end{multline}
\begin{multline}\label{w7-16}
\sum_{n=0}^{\infty}
g_{k n}\, \mathbb{X}^{n} =
\frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \cos (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \cos (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \cos (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} }  = {}\\
{} =
\frac{-1-( f_{k} + h_{k} )\,\mathbb{X}+ {G}_{k}\,\mathbb{X}^2}{1-
h_{k-1}\, \mathbb{X} + \tfrac{1}{2}  \big( h_{k-1}^{2} - h_{2k-1} \big)\, \mathbb{X}^2
-\mathbb{X}^3},
\end{multline}
\begin{multline}
\sum_{n=0}^{\infty}
h_{kn-1}\, \mathbb{X}^{n} =
\frac{1}{1-\big(2\, \cos (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{1}{1-\big(2\, \cos (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{1}{1-\big(2\, \cos (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} }  = {}\\
{} =
\frac{3-2\, h_{k-1}\,\mathbb{X}+ \tfrac{1}{2}  \big( h_{k-1}^{2} - h_{2k-1} \big)\,\,\mathbb{X}^2}{1-
h_{k-1}\, \mathbb{X} + \tfrac{1}{2}  \big( h_{k-1}^{2} - h_{2k-1} \big)\, \mathbb{X}^2
-\mathbb{X}^3}
\stackrel{by~(\ref{wzor-gw1new})}{=}
\frac{3-2\, {\cal B}_{k}\,\mathbb{X}+ (-1)^k\,{\cal A}_{k}\,\mathbb{X}^2}{1-
{\cal B}_{k}\, \mathbb{X} + (-1)^k\,{\cal A}_{k}\, \mathbb{X}^2
-\mathbb{X}^3}.
\end{multline}
($h_{-1}:=1$).

We note that the special case of the formulas~(\ref{w7-15}) and~(\ref{w7-16}) for $k=1$ can be treated
in another way (polynomial~(\ref{w1-d}) will be denoted here  by~$p_7(\mathbb{X})$):
\begin{multline}
\sum_{n=0}^{\infty}
h_{n}\, \mathbb{X}^{n} =
\frac{\xi+\xi^6}{1-(\xi+\xi^6)\, \mathbb{X}} +
\frac{\xi^2+\xi^5}{1-(\xi^2+\xi^5)\, \mathbb{X}} +
\frac{\xi^3+\xi^4}{1-(\xi^3+\xi^4)\, \mathbb{X}} = {}\\
{}=
-\frac{\big( \mathbb{X}^3\,p_{7}\big( 1/\mathbb{X}\big) \big)^{'}}{\mathbb{X}^3\, p_{7}\big( 1/\mathbb{X}\big)}
 = \frac{-1+4\, \mathbb{X}+3\, \mathbb{X}^2}{1+\mathbb{X} -2\,\mathbb{X}^2-\mathbb{X}^3},
\end{multline}
hence by~(\ref{f2s-w3g}):
\begin{multline}
\sum_{n=0}^{\infty}
g_{n}\, \mathbb{X}^{n} =
-1 + \sum_{n=1}^{\infty} \big( h_{n+1} - 2\, h_{n-1}\big)\, \mathbb{X}^{n} = {}\\
{} = -1 + \mathbb{X}^{-1}\, \sum_{n=0}^{\infty} h_{n}\, \mathbb{X}^{n}
-\mathbb{X}^{-1}\, \big( h_{0}+h_{1}\, \mathbb{X}\big) -2\, \mathbb{X}\, \sum_{n=0}^{\infty} h_{n}\, \mathbb{X}^{n} = {}\\
{}=
-6+\mathbb{X}^{-1} +\frac{1-2\, \mathbb{X}^2}{\mathbb{X}}\, \sum_{n=0}^{\infty} h_{n}\, \mathbb{X}^{n} =
\frac{-1-3\, \mathbb{X}+3\, \mathbb{X}^2}{1+\mathbb{X} -2\,\mathbb{X}^2-\mathbb{X}^3}
\end{multline}
and, finally, using~(\ref{f2s-w3g}) again we get
\begin{multline}
\sum_{n=0}^{\infty}
f_{n}\, \mathbb{X}^{n} =
\sum_{n=0}^{\infty} \big( g_{n+1} - h_{n}\big)\, \mathbb{X}^{n} = {}\\
{} = \mathbb{X}^{-1}\, \sum_{n=0}^{\infty} g_{n}\, \mathbb{X}^{n}
+\mathbb{X}^{-1} - \sum_{n=0}^{\infty} h_{n}\, \mathbb{X}^{n} =
-\frac{1+3\, \mathbb{X}+4\, \mathbb{X}^2}{1+\mathbb{X} -2\,\mathbb{X}^2-\mathbb{X}^3}.
\end{multline}





By~(\ref{w1-a}), (\ref{w4.4})--(\ref{w4.6}), (\ref{f2s-tw2-w-f})
and~(\ref{w4.19})--(\ref{w4.21}) we have
\begin{multline}
\sum_{n=0}^{\infty}
x_{k n}\, \mathbb{X}^{n} =
\frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} }  = {}\\
{} =
\frac{\sqrt{7}-( y_{k} + z_{k} )\,\mathbb{X}+ x_{k}^{*}\,\mathbb{X}^2}{1-
z_{k-1}\, \mathbb{X} + \tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2
-\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3},
\end{multline}
\begin{multline}
\sum_{n=0}^{\infty}
y_{k n}\, \mathbb{X}^{n} =
\frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} }  = {}\\
{} =
\frac{\sqrt{7}-( x_{k} + z_{k} )\,\mathbb{X}+ y_{k}^{*}\,\mathbb{X}^2}{1-
z_{k-1}\, \mathbb{X} + \tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2
-\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3},
\end{multline}
\begin{multline}
\sum_{n=0}^{\infty}
z_{k n}\, \mathbb{X}^{n} =
\frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} }  = {}\\
{} =
\frac{\sqrt{7}-( x_{k} + y_{k} )\,\mathbb{X}+ z_{k}^{*}\,\mathbb{X}^2}{1-
z_{k-1}\, \mathbb{X} + \tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2
-\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3},
\end{multline}
and ($z_{-1}:=1$):
\begin{multline}
\sum_{n=0}^{\infty}
z_{k n - 1}\, \mathbb{X}^{n} =
\Big(
1- \big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X}
\Big)^{-1} +
\Big(
1- \big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X}
\Big)^{-1} + {} \\
{} +
\Big(
1- \big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X}
\Big)^{-1}
 =
\frac{3-2\, z_{k-1}\,\mathbb{X}+ \tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\,\mathbb{X}^2}{1-
z_{k-1}\, \mathbb{X} + \tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2
-\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3}.
\end{multline}



By~(\ref{w1-a}), (\ref{w1-d}), (\ref{w4.1})--(\ref{w4.3}),
(\ref{f2s-tw2-w-f}), (\ref{w4.16a})--(\ref{w4.18}),
(\ref{w4.30})--(\ref{w4.32}) and~(\ref{w4.6}), respectively, we get
\begin{multline}
\sum_{n=0}^{\infty}
u_{k n}\, \mathbb{X}^{n} =
\frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} }  = {}\\
{} =
\frac{-1-( v_{k} + w_{k} )\,\mathbb{X}+ u_{k}^{*}\,\mathbb{X}^2}{1-
z_{k-1}\, \mathbb{X} + \tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2
-\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3},
\end{multline}
\begin{multline}
\sum_{n=0}^{\infty}
v_{k n}\, \mathbb{X}^{n} =
\frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} }  = {}\\
{} =
\frac{-1-( u_{k} + w_{k} )\,\mathbb{X}+ v_{k}^{*}\,\mathbb{X}^2}{1-
z_{k-1}\, \mathbb{X} + \tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2
-\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3},
\end{multline}
\begin{multline}
\sum_{n=0}^{\infty}
w_{k n}\, \mathbb{X}^{n} =
\frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{2\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(2\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} }  = {}\\
{} =
\frac{-1-( u_{k} + v_{k} )\,\mathbb{X}+ w_{k}^{*}\,\mathbb{X}^2}{1-
z_{k-1}\, \mathbb{X} + \tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X}^2
-\big({-}\sqrt{7}\,\big)^{k}\, \mathbb{X}^3},
\end{multline}
\begin{multline}
\sum_{n=0}^{\infty}
u_{k n}^{*}\, \mathbb{X}^{n} =
\frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + {}\\
{}+\frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} }  %= {}\\
{} =
\frac{-1-( v_{k}^{*} + w_{k}^{*} )\,\mathbb{X}+
\big({-}\sqrt{7}\,\big)^{k}\, u_{k}\,\mathbb{X}^2}{1-
\tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X} +
\big({-}\sqrt{7}\,\big)^{k}\, z_{k-1}\,\mathbb{X}^2
-7^{k}\, \mathbb{X}^3},
\end{multline}
\begin{multline}
\sum_{n=0}^{\infty}
v_{k n}^{*}\, \mathbb{X}^{n} =
\frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + {}\\
{}+\frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} }  %= {}\\
{} =
\frac{-1-( u_{k}^{*} + w_{k}^{*} )\,\mathbb{X}+
\big({-}\sqrt{7}\,\big)^{k}\, v_{k}\,\mathbb{X}^2}{1-
\tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X} +
\big({-}\sqrt{7}\,\big)^{k}\, z_{k-1}\,\mathbb{X}^2
-7^{k}\, \mathbb{X}^3},
\end{multline}
\begin{multline}
\sum_{n=0}^{\infty}
w_{k n}^{*}\, \mathbb{X}^{n} =
\frac{2\, \cos \big(\frac{2\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \cos \big(\frac{4\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } + {}\\
{}+\frac{2\, \cos \big(\frac{8\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} }  %= {}\\
{} =
\frac{-1-( u_{k}^{*} + v_{k}^{*} )\,\mathbb{X}+
\big({-}\sqrt{7}\,\big)^{k}\, w_{k}\,\mathbb{X}^2}{1-
\tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X} +
\big({-}\sqrt{7}\,\big)^{k}\, z_{k-1}\,\mathbb{X}^2
-7^{k}\, \mathbb{X}^3}.
\end{multline}
By~(\ref{w1-a}), (\ref{w4.19})--(\ref{w4.21}), (\ref{w4.33})--(\ref{w4.35})
and~(\ref{w4.6}), we obtain
\begin{multline}
\sum_{n=0}^{\infty}
x_{k n}^{*}\, \mathbb{X}^{n} =
\frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + {}\\
{}+\frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} }  %= {}\\
{} =
\frac{\sqrt{7}-( y_{k}^{*} + z_{k}^{*} )\,\mathbb{X}+
\big({-}\sqrt{7}\,\big)^{k}\, x_{k}\,\mathbb{X}^2}{1-
\tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X} +
\big({-}\sqrt{7}\,\big)^{k}\, z_{k-1}\,\mathbb{X}^2
-7^{k}\, \mathbb{X}^3},
\end{multline}
\begin{multline}
\sum_{n=0}^{\infty}
y_{k n}^{*}\, \mathbb{X}^{n} =
\frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} } + {}\\
{}+\frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} }  %= {}\\
{} =
\frac{\sqrt{7}-( x_{k}^{*} + z_{k}^{*} )\,\mathbb{X}+
\big({-}\sqrt{7}\,\big)^{k}\, y_{k}\,\mathbb{X}^2}{1-
\tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X} +
\big({-}\sqrt{7}\,\big)^{k}\, z_{k-1}\,\mathbb{X}^2
-7^{k}\, \mathbb{X}^3},
\end{multline}
\begin{multline}
\sum_{n=0}^{\infty}
z_{k n}^{*}\, \mathbb{X}^{n} =
\frac{2\, \sin \big(\frac{2\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{4\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } +
\frac{2\, \sin \big(\frac{4\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{8\, \pi}{7}) \big)^{k}\, \mathbb{X} } + {}\\
{}+\frac{2\, \sin \big(\frac{8\, \pi}{7}\big)}{1-\big(4\, \sin (\tfrac{2\, \pi}{7})\, \sin (\tfrac{4\, \pi}{7}) \big)^{k}\, \mathbb{X} }  %= {}\\
{} =
\frac{\sqrt{7}-( x_{k}^{*} + y_{k}^{*} )\,\mathbb{X}+
\big({-}\sqrt{7}\,\big)^{k}\, z_{k}\,\mathbb{X}^2}{1-
\tfrac{1}{2}  \big( z_{k-1}^{2} - z_{2k-1} \big)\, \mathbb{X} +
\big({-}\sqrt{7}\,\big)^{k}\, z_{k-1}\,\mathbb{X}^2
-7^{k}\, \mathbb{X}^3}.
\end{multline}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%               %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%   Section 8   %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%               %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Jordan decomposition}

For sequences $A_{n}(\delta)$, $B_{n}(\delta)$ and $C_{n}(\delta)$
we have the equality
\begin{equation}
\left[
\begin{array}{l}
A_{n+1}(\delta) \\
B_{n+1}(\delta) \\
C_{n+1}(\delta) \\
\end{array}
\right]
=
\mathcal{W}(\delta)
\left[
\begin{array}{l}
A_{n}(\delta) \\
B_{n}(\delta) \\
C_{n}(\delta) \\
\end{array}
\right],
\qquad n\in \mathbb{N},
\end{equation}
where
\begin{equation}
\mathcal{W}(\delta)=
\left[
\begin{array}{ccc}
1 & 2\, \delta & -\delta \\
\delta & 1 & 0 \\
0 & \delta & 1-\delta \\
\end{array}
\right].
\end{equation}
Matrix $\mathcal{W}(\delta)$ is a~diagonalized matrix,
and the following decomposition can be obtained
\begin{equation}
\mathcal{W}(\delta)=
A\cdot
\left[
\begin{array}{ccc}
1+\delta\, (\xi+\xi^6) & 0 & 0 \\
0 & 1+\delta\, (\xi^2+\xi^5) & 0 \\
0 & 0 & 1+\delta\, (\xi^3+\xi^4) \\
\end{array}
\right]
\cdot A^{-1},
\end{equation}
where
\begin{equation*}
A =
\left[
\begin{array}{ccc}
1+(\xi+\xi^6)^{-1} & 1+(\xi^2+\xi^5)^{-1} & 1+(\xi^3+\xi^4)^{-1} \\[1ex]
(\xi+\xi^6)^{-2}+(\xi+\xi^6)^{-1} & (\xi^2+\xi^5)^{-2}+(\xi^2+\xi^5)^{-1} & (\xi^3+\xi^4)^{-2}+(\xi^3+\xi^4)^{-1} \\[1ex]
(\xi+\xi^6)^{-2} & (\xi^2+\xi^5)^{-2} & (\xi^3+\xi^4)^{-2} \\
\end{array}
\right]
\end{equation*}
and
\begin{equation*}
A^{-1} = \frac{1}{7}
\left[
\begin{array}{ccc}
(\xi^{\phantom{2}}+\xi^6)^{2}   & (\xi^{\phantom{2}}+\xi^6)^{3}   & (\xi^{\phantom{2}}+\xi^6)^{4}  \\[1ex]
(\xi^2+\xi^5)^{2} & (\xi^2+\xi^5)^{3} & (\xi^2+\xi^5)^{4}\\[1ex]
(\xi^3+\xi^4)^{2} & (\xi^3+\xi^4)^{3} & (\xi^3+\xi^4)^{4}\\
\end{array}
\right]    \cdot
\left[
\begin{array}{ccc}
-3 & 2 & 5 \\[1ex]
1 & 1 & -2 \\[1ex]
2 & -1 & -2 \\
\end{array}
\right].
\end{equation*}

It should be noticed, that characteristic polynomial $w(\lambda)$
of matrix $\mathcal{W}(\delta)$ is equal to
$$
w(\lambda) = -\delta^3\, p_{7} \Big( \frac{\lambda-1}{\delta} \Big).
$$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Tables}

\renewcommand{\arraystretch}{1.3}


\begin{table}[H]
\begin{center}

{\small

\caption{}\label{f2s-tab1}
\medskip

\begin{tabular}{||c||r|r|r||r|r|r||r|r|r||}\hline\hline
$n$ &
\multicolumn{1}{|c|}{$a_{n}/\sqrt{7}$} &
\multicolumn{1}{|c|}{$b_{n}/\sqrt{7}$} &
\multicolumn{1}{|c||}{$c_{n}/\sqrt{7}$} &
\multicolumn{1}{|c|}{$\alpha_{n}/\sqrt{7}$} &
\multicolumn{1}{|c|}{$\beta_{n}/\sqrt{7}$} &
\multicolumn{1}{|c||}{$\gamma_{n}/\sqrt{7}$} &
\multicolumn{1}{|c|}{$f_{n}$} &
\multicolumn{1}{|c|}{$g_{n}$} &
\multicolumn{1}{|c||}{$h_{n}$} \\ \hline\hline
$0 $ & $1     $ & $1     $ & $1      $ & $0      $ & $-2     $ & $1     $ & $-1  $  & $-1  $  & $-1  $ \\ \hline
$1 $ & $3     $ & $2     $ & $0      $ & $-2     $ & $-5     $ & $3     $ & $-2  $  & $-2  $  & $5   $ \\ \hline
$2 $ & $8     $ & $7     $ & $-2     $ & $-9     $ & $-15    $ & $8     $ & $-4  $  & $3   $  & $-4  $ \\ \hline
$3 $ & $23    $ & $24    $ & $-9     $ & $-33    $ & $-47    $ & $23    $ & $-1  $  & $-8  $  & $13  $ \\ \hline
$4 $ & $70    $ & $80    $ & $-33    $ & $-113   $ & $-150   $ & $70    $ & $-9  $  & $12  $  & $-16 $ \\ \hline
$5 $ & $220   $ & $263   $ & $-113   $ & $-376   $ & $-483   $ & $220   $ & $3   $  & $-25 $  & $38  $ \\ \hline
$6 $ & $703   $ & $859   $ & $-376   $ & $-1235  $ & $-1562  $ & $703   $ & $-22 $  & $41  $  & $-57 $ \\ \hline
$7 $ & $2265  $ & $2797  $ & $-1235  $ & $-4032  $ & $-5062  $ & $2265  $ & $19  $  & $-79 $  & $117 $ \\ \hline
$8 $ & $7327  $ & $9094  $ & $-4032  $ & $-13126 $ & $-16421 $ & $7327  $ & $-60 $  & $136 $  & $-193$ \\ \hline
$9 $ & $23748 $ & $29547 $ & $-13126 $ & $-42673 $ & $-53295 $ & $23748 $ & $76  $  & $-253$  & $370 $ \\ \hline
$10$ & $77043 $ & $95968 $ & $-42673 $ & $-138641$ & $-173011$ & $77043 $ & $-177$  & $446 $  & $-639$ \\ \hline
$11$ & $250054$ & $311652$ & $-138641$ & $-450293$ & $-561706$ & $250054$ & $269 $  & $-816$  & $1186$ \\ \hline\hline
\end{tabular}
}
\end{center}

\end{table}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{table}[H]


\begin{center}
{\small

\caption{}\label{f2s-tab7}
\medskip

\begin{tabular}{||c||r|r|r|r|r|r|r|r|r|r|r|r||}\hline\hline
$n$  &
$0 $ & $1 $ & $2 $ & $3 $ & $4 $ & $5 $ & $6 $ & $7 $ & $8 $ & $9 $ & $10$ & $11$ \\ \hline\hline
$\varepsilon_{n}$ &
$1  $ & $0  $ & $-2 $ & $2  $ & $2  $ & $-6 $ & $2  $ & $10 $ & $-14$ & $-6 $ & $34 $ & $-22$ \\ \hline
$\omega_{n}$ &
$0  $ & $1  $ & $-1 $ & $-1 $ & $3  $ & $-1 $ & $-5 $ & $7  $ & $3  $ & $-17$ & $11 $ & $23 $ \\ \hline
$\Psi_{n}$ &
$2   $ & $-1  $ & $-3  $ & $5   $ & $1   $ & $-11 $ & $9   $ & $13  $ & $-31 $ & $5   $ & $57  $ & $-67 $ \\ \hline\hline
\end{tabular}
}
\end{center}

\end{table}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[H]


\begin{center}
{\small

\caption{}\label{f2s-tab3}
\medskip

\begin{tabular}{||c||r|r|r||r|r|r||}\hline\hline
$n$  &
\multicolumn{1}{|c|}{$u_{n}$} &
\multicolumn{1}{|c|}{$v_{n}$} &
\multicolumn{1}{|c||}{$w_{n}$} &
\multicolumn{1}{|c|}{$x_{n}$} &
\multicolumn{1}{|c|}{$y_{n}$} &
\multicolumn{1}{|c||}{$z_{n}$} \\ \hline\hline
$0 $ &  $-1           $ & $-1            $ & $-1          $ & $\sqrt{7}     $ & $\sqrt{7}    $ & $\sqrt{7}    $ \\ \hline
$1 $ &  $\sqrt{7}     $ & $-2 \sqrt{7}   $ & $0           $ & $0            $ & $0           $ & $7           $ \\ \hline
$2 $ &  $0            $ & $-7            $ & $0           $ & $\sqrt{7}     $ & $2 \sqrt{7}  $ & $4 \sqrt{7}  $ \\ \hline
$3 $ &  $\sqrt{7}     $ & $-6 \sqrt{7}   $ & $\sqrt{7}    $ & $0            $ & $7           $ & $21          $ \\ \hline
$4 $ &  $0            $ & $-28           $ & $7           $ & $0            $ & $7 \sqrt{7}  $ & $14 \sqrt{7} $ \\ \hline
$5 $ &  $0            $ & $-21 \sqrt{7}  $ & $7 \sqrt{7}  $ & $-7           $ & $35          $ & $70          $ \\ \hline
$6 $ &  $-7           $ & $-105          $ & $42          $ & $-7 \sqrt{7}  $ & $28 \sqrt{7} $ & $49 \sqrt{7} $ \\ \hline
$7 $ &  $-7 \sqrt{7}  $ & $-77 \sqrt{7}  $ & $35 \sqrt{7} $ & $-49          $ & $147         $ & $245         $ \\ \hline
$8 $ &  $-49          $ & $-392          $ & $196         $ & $-42 \sqrt{7} $ & $112 \sqrt{7}$ & $175 \sqrt{7}$ \\ \hline
$9 $ &  $-42 \sqrt{7} $ & $-287 \sqrt{7} $ & $154 \sqrt{7}$ & $-245         $ & $588         $ & $882         $ \\ \hline
$10$ &  $-245         $ & $-1470         $ & $833         $ & $-196 \sqrt{7}$ & $441 \sqrt{7}$ & $637 \sqrt{7}$ \\ \hline
$11$ &  $-196 \sqrt{7}$ & $-1078 \sqrt{7}$ & $637 \sqrt{7}$ & $-1078        $ & $2303        $ & $3234        $ \\ \hline\hline
\end{tabular}
}
\end{center}

\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[H]


\begin{center}
{\small

\caption{}\label{f2s-tab2}
\medskip

\begin{tabular}{||c||r|r|r||r|r|r||}\hline\hline
$n$  &
\multicolumn{1}{|c|}{$u_{n}^{*}$} &
\multicolumn{1}{|c|}{$v_{n}^{*}$} &
\multicolumn{1}{|c||}{$w_{n}^{*}$} &
\multicolumn{1}{|c|}{$x_{n}^{*}/\sqrt{7}$} &
\multicolumn{1}{|c|}{$y_{n}^{*}/\sqrt{7}$} &
\multicolumn{1}{|c||}{$z_{n}^{*}/\sqrt{7}$} \\ \hline\hline
$0 $ &  $-1     $ & $-1    $ & $-1    $ & $1       $ & $1       $ & $1       $ \\ \hline
$1 $ &  $-7     $ & $7     $ & $0     $ & $1       $ & $2       $ & $-3      $ \\ \hline
$2 $ &  $-14    $ & $7     $ & $-7    $ & $7       $ & $7       $ & $0       $ \\ \hline
$3 $ &  $-56    $ & $42    $ & $-7    $ & $14      $ & $21      $ & $-14     $ \\ \hline
$4 $ &  $-147   $ & $98    $ & $-49   $ & $56      $ & $63      $ & $-21     $ \\ \hline
$5 $ &  $-490   $ & $343   $ & $-98   $ & $147     $ & $196     $ & $-98     $ \\ \hline
$6 $ &  $-1421  $ & $980   $ & $-392  $ & $490     $ & $588     $ & $-245    $ \\ \hline
$7 $ &  $-4459  $ & $3087  $ & $-1029 $ & $1421    $ & $1813    $ & $-833    $ \\ \hline
$8 $ &  $-13377 $ & $9261  $ & $-3430 $ & $4459    $ & $5488    $ & $-2401   $ \\ \hline
$9 $ &  $-41160 $ & $28469 $ & $-9947 $ & $13377   $ & $16807   $ & $-7546   $ \\ \hline
$10$ &  $-124852$ & $86436 $ & $-31213$ & $41160   $ & $51107   $ & $-22638  $ \\ \hline
$11$ &  $-381759$ & $264110$ & $-93639$ & $124852  $ & $156065  $ & $-69629  $ \\ \hline\hline
\end{tabular}
}
\end{center}

\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[H]


\begin{center}
{\small

\caption{}\label{f2s-tab6}
\medskip

\begin{tabular}{||c||r|r|r||r|r|r||}\hline\hline
$n$  &
\multicolumn{1}{|c|}{$F_{n}$} &
\multicolumn{1}{|c|}{$G_{n}$} &
\multicolumn{1}{|c||}{$H_{n}$} &
\multicolumn{1}{|c|}{$\widetilde{A}_{n}/\sqrt{7}$} &
\multicolumn{1}{|c|}{$\widetilde{B}_{n}/\sqrt{7}$} &
\multicolumn{1}{|c||}{$\widetilde{C}_{n}/\sqrt{7}$} \\ \hline\hline
$0 $ &  $-1   $ & $-1   $ & $-1   $ & $1          $ & $1         $ & $1         $ \\ \hline
$1 $ &  $-4   $ & $3    $ & $3    $ & $-1         $ & $3         $ & $4         $ \\ \hline
$2 $ &  $5    $ & $-9   $ & $-2   $ & $-8         $ & $15        $ & $19        $ \\ \hline
$3 $ &  $-15  $ & $20   $ & $6    $ & $-42        $ & $76        $ & $95        $ \\ \hline
$4 $ &  $31   $ & $-46  $ & $-11  $ & $-213       $ & $384       $ & $479       $ \\ \hline
$5 $ &  $-72  $ & $103  $ & $26   $ & $-1076      $ & $1939      $ & $2418      $ \\ \hline
$6 $ &  $160  $ & $-232 $ & $-57  $ & $-5433      $ & $9790      $ & $12208     $ \\ \hline
$7 $ &  $-361 $ & $521  $ & $129  $ & $-27431     $ & $49429     $ & $61637     $ \\ \hline
$8 $ &  $810  $ & $-1171$ & $-289 $ & $-138497    $ & $249563    $ & $311200    $ \\ \hline
$9 $ &  $-1821$ & $2631 $ & $650  $ & $-699260    $ & $1260023   $ & $1571223   $ \\ \hline
$10$ &  $4091 $ & $-5912$ & $-1460$ & $-3530506   $ & $6361752   $ & $7932975   $ \\ \hline
$11$ &  $-9193$ & $13284$ & $3281 $ & $-17825233  $ & $32119960  $ & $40052935  $ \\ \hline\hline
\end{tabular}
}
\end{center}

\end{table}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\renewcommand{\arraystretch}{1.3}

\begin{table}[H]
{\small %scriptsize
\caption{}\label{f2s-tab5}
\medskip

\mbox{\null}\hfill
\begin{tabular}{||c||l||} \hline\hline
$n$      & \multicolumn{1}{|c||}{$\Omega_n(\delta )$} \\ \hline\hline
$0$ & $3 $ \\ \hline
$1$ & $i \sqrt{7} \delta +3                                                                                                                $ \\ \hline
$2$ & $-7 \delta^2+2 i \sqrt{7} \delta +3                                                                                                  $ \\ \hline
$3$ & $-4 i \sqrt{7} \delta^3-21 \delta^2+3 i \sqrt{7}\delta +3                                                                            $ \\ \hline
$4$ & $21 \delta^4-16 i \sqrt{7} \delta^3-42 \delta^2+4 i \sqrt{7} \delta +3                                                               $ \\ \hline
$5$ & $14 i \sqrt{7} \delta^5+105 \delta^4-40 i \sqrt{7} \delta^3-70 \delta^2+5 i \sqrt{7} \delta +3                                       $ \\ \hline
$6$ & $-70 \delta^6+84 i \sqrt{7} \delta^5+315 \delta^4-80 i \sqrt{7} \delta^3-105\delta^2+6 i \sqrt{7} \delta +3                          $ \\ \hline
$7$ & $-49 i \sqrt{7} \delta^7-490 \delta^6+294 i \sqrt{7} \delta^5+735 \delta^4-140 i\sqrt{7} \delta^3-147 \delta^2+7 i\sqrt{7} \delta +3 $ \\ \hline\hline
\end{tabular}
\hfill\mbox{\null}

}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\renewcommand{\arraystretch}{1.3}

\begin{table}[H]
{\scriptsize
\caption{}\label{f2s-tab4}
\medskip

\mbox{\null}\hfill
\begin{tabular}{||c|l|l||} \hline\hline
$n$      & \multicolumn{2}{|c||}{$p_n(\delta )$} \\ \hline\hline
$0$ & \multicolumn{2}{|l||}{$1 $} \\ \hline
$1$ & \multicolumn{2}{|l||}{$1                                                                                                            $} \\ \hline
$2$ & \multicolumn{2}{|l||}{$-2 \delta^2 + 1                                                                                              $} \\ \hline
$3$ & \multicolumn{2}{|l||}{$-i \sqrt{7} \delta^3-6\delta^2+1                                                                            $} \\ \hline
$4$ & \multicolumn{2}{|l||}{$3 \delta^4-4 i \sqrt{7} \delta^3-12 \delta ^2+1                                                              $} \\ \hline
$5$ & \multicolumn{2}{|l||}{$5 i \sqrt{7} \delta^5+15   \delta^4-10 i \sqrt{7} \delta^3-20 \delta^2+1                                     $} \\ \hline
$6$ & \multicolumn{2}{|l||}{$-8 \delta^6+30 i \sqrt{7} \delta^5+45 \delta^4-20 i \sqrt{7} \delta^3-30 \delta^2+1                          $} \\ \hline
$7$ & \multicolumn{2}{|l||}{$-17 i \sqrt{7} \delta^7-56 \delta^6+105 i  \sqrt{7} \delta^5+105 \delta^4-35 i \sqrt{7}\delta^3-42 \delta^2+1$} \\ \hline\hline
$n$      &  \multicolumn{2}{|c||}{$r_n(\delta )$}      \\ \hline\hline
$0$ & \multicolumn{2}{|l||}{$0 $} \\ \hline
$1$ &  \multicolumn{2}{|l||}{$\delta                                                                                    $} \\ \hline
$2$ &  \multicolumn{2}{|l||}{$2 \delta                                                                                  $} \\ \hline
$3$ &  \multicolumn{2}{|l||}{$-2 \delta^3+3 \delta                                                                      $} \\ \hline
$4$ &  \multicolumn{2}{|l||}{$-i \sqrt{7} \delta^4-8 \delta^3+4 \delta                                                  $} \\ \hline
$5$ &  \multicolumn{2}{|l||}{$3 \delta^5-5 i \sqrt{7} \delta^4-20 \delta^3+5 \delta                                     $} \\ \hline
$6$ &  \multicolumn{2}{|l||}{$5 i \sqrt{7} \delta^6+18 \delta^5-15 i \sqrt{7} \delta^4-40 \delta^3+6 \delta             $} \\ \hline
$7$ &  \multicolumn{2}{|l||}{$-8 \delta^7+35 i \sqrt{7} \delta^6+63 \delta^5-35 i \sqrt{7} \delta^4-70 \delta^3+7 \delta$} \\ \hline\hline
 $n$     &  \multicolumn{1}{|c|}{$s_n(\delta )$}  & \multicolumn{1}{|c||}{$k_n(\delta )$} \\ \hline\hline
$0$ &  $0 $ & $0$ \\ \hline
$1$ &  $0                                                         $ &  $0                                            $ \\ \hline
$2$ &  $0                                                         $ &  $0                                            $ \\ \hline
$3$ &  $\delta^3                                                  $ &  $0                                            $ \\ \hline
$4$ &  $4 \delta^3                                                $ &  $-2 \delta^4                                  $ \\ \hline
$5$ &  $-5 \delta^5+10 \delta^3                                   $ &  $-10 \delta^4                                 $ \\ \hline
$6$ &  $-i \sqrt{7} \delta^6-30 \delta^5+20 \delta^3              $ &  $10 \delta^6-30 \delta^4                      $ \\ \hline
$7$ &  $18 \delta^7-7 i \sqrt{7} \delta^6-105 \delta^5+35 \delta^3$ &  $2 i \sqrt{7} \delta^7+70 \delta^6-70 \delta^4$ \\ \hline\hline
$n$      &  \multicolumn{2}{|c||}{$l_n(\delta )$} \\ \hline\hline
$0$ & \multicolumn{2}{|l||}{$0 $} \\ \hline
$1$ &  \multicolumn{2}{|l||}{$0                                                                                $} \\ \hline
$2$ &  \multicolumn{2}{|l||}{$\delta^2                                                                         $} \\ \hline
$3$ &  \multicolumn{2}{|l||}{$3 \delta^2                                                                       $} \\ \hline
$4$ &  \multicolumn{2}{|l||}{$-3 \delta^4 + 6 \delta^2                                                         $} \\ \hline
$5$ &  \multicolumn{2}{|l||}{$-i \sqrt{7} \delta^5-15 \delta^4+10 \delta^2                                     $} \\ \hline
$6$ &  \multicolumn{2}{|l||}{$8 \delta^6-6 i \sqrt{7} \delta^5-45 \delta^4+15 \delta^2                         $} \\ \hline
$7$ &  \multicolumn{2}{|l||}{$6 i \sqrt{7} \delta^7+56 \delta^6-21 i \sqrt{7} \delta^5-105 \delta^4+21 \delta^2$} \\ \hline\hline
\end{tabular}
\hfill\mbox{\null}

}
\end{table}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{Acknowledgments}

The authors wish to express their gratitude to the referee
for several helpful comments and suggestions
concerning the first version of our paper,
especially for preparing Example~\ref{example2.2}, which is
added to the present version of the paper.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{thebibliography}


\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}:
Primary 11B83, 11A07; Secondary 39A10.

\noindent \emph{Keywords: } Ramanujan's formulas,
Fibonacci numbers, primitive roots of unity, recurrence relation.



\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A001607}, \seqnum{A002249}, \seqnum{A079309}, \seqnum{A094648}, \seqnum{A110512}, and
\seqnum{A115146}.)

\bigskip
\hrule
\bigskip


\vspace*{+.1in}
\noindent
Received April 19 2007;
revised version received May 17 2007.
Published in {\it Journal of Integer Sequences}, June 4 2007.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}.
\vskip .1in




\end{document}
