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\begin{center}
	\vskip 1cm{\LARGE\bf 
		Ellipse Chains and Associated Sequences
	}
	\vskip 1cm
	\large
	Hac\`ene Belbachir and Soumeya Merwa Tebtoub \\
	Department of Mathematics\\
	USTHB, RECITS Laboratory\\
	P. O. Box 32 \\
	El Alia 16111 \\
	Bab Ezzouar \\
	Algiers \\
	Algeria\\
	\href{mailto:hbelbachir@usthb.dz}{\tt hbelbachir@usthb.dz},\ 
	\href{mailto:hacenebelbachir@gmail.com}{\tt hacenebelbachir@gmail.com}\\ \href{mailto:stebtoub@usthb.dz}{\tt stebtoub@usthb.dz}, \ 
	\href{mailto:tebtoubsoumeya@gmail.com}{\tt tebtoubsoumeya@gmail.com} \\
        \ \\
	L\'aszl\'o N\'emeth \\
	Institute of Mathematics\\
	University of Sopron\\
	Bajcsy Zs.\ u.\ 4 \\
	Sopron H9400 \\
	Hungary\\ 
	and \\
	USTHB, RECITS Laboratory \\
	Algeria \\
	\href{mailto:nemeth.laszlo@uni-sopron.hu}{\tt nemeth.laszlo@uni-sopron.hu}
	
	
\end{center}

\vskip .2 in
\begin{abstract}
	We define circle and ellipse chains tangent to the branches of a hyperbola and the terms of the chains are mutually tangent to each other. Our goal is to derive recurrence relations for the parameters of chains elements and to establish some connections between integer sequences and chains.
\end{abstract}

\section{Introduction}

Let us consider the hyperbola $\mathcal{H}$ with the canonical equation 
\begin{equation}\label{eq:hiperbola}
	\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1,
\end{equation}
and foci $(\pm c,0)$,
where $a$ and $b$ are positive real numbers and $c^2=a^2+b^2$. Lucca \cite{Lucca} examined a tangential chain of circles inside the  branch $x>0$ of the hyperbola so that the $n$-th circle with center $(x_n,0)$ and radius $r_n$ are tangent to the hyperbola and mutually tangent to each other. He showed that 
for certain $a$ and $b$, the sequences $(x_n/x_0)_{n=0}^{\infty}$ and $(r_n/r_0)_{n=0}^{\infty}$ are integers.   
Belbachir et al.~\cite{BNT} extended the circle chains to ellipse chains inside the branch of hyperbola when the ratio of the minor and major axes is fixed. They described the recurrence relations for the ellipses'  parameters and determined a connection between the parameters of the ellipse chains and integer sequences.

In the present paper,  we give an extension of the papers \cite{BNT, Lucca}. We examine  special  chains of circles and ellipses between the branches of hyperbola $\mathcal{H}$ (or outside $\mathcal{H}$), such that the circles (ellipses) are tangent to the hyperbola $\mathcal{H}$ and mutually tangent to each other.
Furthermore, we give the recurrence relations for the tangential points. 
We define a tangential chain of ellipses  between the branches of $\mathcal{H}$ where the centers of the ellipses coincide with the centers of the circles. We give recurrence relations for the ellipses' parameters.

Our other main purpose is to give integer sequences that describe the
parameters of our chains. We find more than fifty such integer  sequences
that appear in the {\it On-Line Encyclopedia of Integer Sequences} (OEIS) \cite{OEIS}, and in this way, our investigation gives
them geometrical interpretations.  In what follows, we define $t:=a/b$, $s:=b/a$,
$\lambda:=2t^2+1$, and $\mu:=t^2+1$.

\section{Circle chains between the branches of hyperbola}
Let us define a chain of circles with the following properties.

 The canonical equation of the $n$-th circle centered at point $(0,y_n)$, ($n\geq0$, $y_n\geq0$) is  
	\begin{equation}\label{eq:circle}
	x^{2}+(y-y_{n})^{2} =r^2_n,
	\end{equation}
where the center of each circle lies on the $y$-axis, and
$r_n>0$ is the radius (Figure~\ref{fig:ellchan}).

 The circles \eqref{eq:circle} are tangent to the hyperbola
	\eqref{eq:hiperbola} and are mutually tangent, i.e., 
	\begin{equation*}\label{eq:yn-yn}
	y_{n}-y_{n-1}=r_{n}+r_{n-1} \quad (n\geq1).
	\end{equation*}	
Let $(\hat{x}_n,\hat{y}_n)$ be the tangential point of the branch $x>0$ of hyperbola $\mathcal{H}$ and the $n$-th circle of the chain, given by 
$\hat{y}_n = s^2/(1+s^2)\ y_n$ and  $\hat{x}^2_n = 1/(t^2+s^2+2)\ y^2_n +a^2$.
Moreover 
\begin{equation}\label{e:aya}
\hat{x}^2_n = t^2\,\hat{y}_n^2 +a^2. 
\end{equation}
 The sequences $(y_n)_{n\geq0}$, $(r_n)_{n\geq0}$, $(\hat{x}_n)_{n\geq0}$ , and $(\hat{y}_n)_{n\geq0}$ satisfy the  following recurrence relations:
\begin{theorem}\label{th:re_eq_01}
	The sequences $(y_n)_{n\geq0}$, $(r_n)_{n\geq0}$, and $(\hat{y}_n)_{n\geq0}$ are second-order linear homogeneous  recurrence sequences 
	\begin{equation}\label{eq:recurbi}
     \ell_n=2\lambda\ell_{n-1}-\ell_{n-2} \qquad (n\ge2),
    \end{equation} 
    and the sequence $(\hat{x}^2_n)_{n\geq0}$ is a third-order linear homogeneous  recurrence sequence
\begin{equation}\label{e:xk}
	\hat{x}^2_{n}=	(4\lambda^2-1)\,\hat{x}^2_{n-1}- (4\lambda^2-1)\, \hat{x}^2_{n-2}+\hat{x}^2_{n-3} \qquad (n\ge3),
\end{equation} 
	 and the initial values are $y_0=0$, $r_0=a$,  $\hat{x}^2_0=a^2$, $\hat{y}_0=0$, $y_1=2a\mu$, $r_1=a\lambda$, $\hat{x}^2_1=a^2(4t^2+1)$, $\hat{y}_1=2a$, and $\hat{x}^2_2= a^2(16t^2\lambda^2+1)$.
\end{theorem}

\begin{proof}
The system composed of the equations \eqref{eq:hiperbola} and \eqref{eq:circle} gives 
\begin{equation*}
\left\{
\begin{array}{ll} 
y_n=\lambda\, y_{n-1}+2\mu \,r_{n-1}, \\
r_n=2t^2\, y_{n-1}+\lambda\,r_{n-1}.
\end{array}
\right.
\end{equation*}
Then $(y_n)_{n\geq0}$ and $(r_n)_{n\geq0}$ satisfy \eqref{eq:recurbi} and this
is also the case for $(\hat{y}_n)_{n\geq0}$, as $y_n= \mu\,\hat{y}_n$. 
 Now for  $\hat{x}^2_n$,  let $C=1/(t^2+s^2+2)$.  Then $\hat{x}^2_n = C y^2_n +a^2$, 
$y_n= 2\lambda y_{n-1}-y_{n-2}$, substituting $y_n$ in $\hat{x}^2_n$ we get $\hat{x}^2_n = C (2\lambda y_{n-1}-y_{n-2})^2 +a^2 =4\lambda^2 \hat{x}^2_{n-1}+\hat{x}^2_{n-2}-4C\lambda y_{n-1}y_{n-2} -4\lambda^2 a^2$ and from the sum $\hat{x}^2_{n}+\hat{x}^2_{n-1}$, we obtain the equation \eqref{e:xk}.
\end{proof}
\begin{remark}
	Because of the equation \eqref{e:aya}, the recurrence relation 
	for the sequence $(\hat{y}^2_n)_{n\geq0}$ is the same as for the recurrence of $(\hat{x}^2_n)_{n\geq0}$. Thus, the sequence of squared distances of tangential points and the origin $(d^2_n=\hat{x}^2_n+\hat{y}^2_n)_{n\geq0}$ satisfies the recurrence \eqref{e:xk} as well.  In the last section, we give a second-order recurrence solving \eqref{e:xk}. 
  \end{remark}

\section{Ellipse chains between the branches of hyperbola}
In this section, we define a tangential chain of ellipses between the branches of $\mathcal{H}$, where the ellipses' centers coincide with the circles' centers. Let us define a chain of ellipses with the following properties.

The canonical equation of the $n$-th ellipse centered at point $(0,y_n)$ ($y_n>0$, $n\geq0$) is  
\begin{equation}\label{eq:ellipse}
	\frac{x^{2}}{\alpha_n^{2}}+\frac{(y-y_n)^{2}}{\beta_n^{2}}=1,
\end{equation} where $2\alpha_n>0$ is the width and $2\beta_n>0$ is the height of the ellipse  (Figure~\ref{fig:ellchan}).
\begin{figure}[H]
 	\begin{center}
 	\includegraphics{ellipsechain_y01}
	\caption{Ellipse and circle chains between the branches of hyperbola.}
	\label{fig:ellchan}
\end{center}
\end{figure}
Let the equation of the $0$-th ellipse be $x^{2}/a^{2}+y^{2}/\beta_0^{2}=1$,
and thus $y_0=0$, $\alpha_0=a$. This implies that $0\beta_0<r_0+r_1=2a\mu$.
 
 The ellipses \eqref{eq:ellipse} are tangent to the hyperbola
	\eqref{eq:hiperbola} and are mutually tangent, i.e.,
	\begin{equation}\label{eq:ellyn-yn}
	y_{n}-y_{n-1}=\beta_{n}+\beta_{n-1}  \quad (n\geq1).
	\end{equation} 
The sequence $y_n$ is the same as the sequence of circle chains. However, we can give recurrence formulas for the parameter $\beta_n$ of ellipses \eqref{recbeta}. Let us determine $\alpha_{n}^2$. From the system composed by the equations \eqref{eq:hiperbola} and \eqref{eq:ellipse} and from the tangency condition between the ellipses, we have
\begin{equation}\label{eq:el1}
\left(\dfrac{a^2}{b^2}+\dfrac{\alpha_{n}^2}{\beta_{n}^2}\right)y^2-\dfrac{2\alpha_{n}^2}{\beta_{n}^2}y_{n}y+\dfrac{\alpha_{n}^{2}}{\beta_{n}^2}y_{n}^2+a^2-\alpha_{n}^2=0.
\end{equation}
 Since the discriminant of equation \eqref{eq:el1} is equal to zero and after simplification, we get
\begin{equation}\label{t}
s^2\alpha_{n}^{4}-\left(b^{2}+y_{n}^{2}-\beta_{n}^{2}\right)\alpha_{n}^{2}-a^{2}\beta_{n}^2=0.
\end{equation}
We put $\delta_{n}=\alpha_{n}^{2}$ and $\omega_{n}=b^{2}+y_{n}^{2}-\beta_{n}^{2}$. Then the solutions of the equation \eqref{t} are 
 \hbox{$\delta_{n,1}=(\omega_{n}+\sqrt{\omega_{n}^{2}+4b^{2}\beta_{n}^{2}})/2s^2$} and 
$\delta_{n,2}=(\omega_{n}-\sqrt{\omega_{n}^{2}+4b^{2}\beta_{n}^{2}})/2s^2$.
Since $\delta_{n,2}<0$, we get 
\hbox{$ \alpha^2_{n}=t^2/2(\omega_{n}+\sqrt{\omega_{n}^{2}+4b^{2}\beta_{n}^{2}})$.}
Let $(\tilde{x}_n,\tilde{y}_n)$ be the tangential point of the branch $x>0$ of $\mathcal{H}$ and the $n$-th ellipse of the chain. Then recurrence relations
for $\tilde{x}_n $ and $\tilde{y}_n$
are as follows: $\tilde{y}_{n}=({\alpha_{n}^2b^2}/{a^2\beta_{n}^2+\alpha_{n}^2b^2})y_{n}$ and $\tilde{x}^2_{n}=({\alpha_{n}^4a^2b^2}/{(a^2\beta_{n}^2+\alpha_{n}^2b^2)^2})y_{n}^2+a^2$,
where $y^2_n=(\alpha_n^2-a^2)(a^2\beta_{n}^2+\alpha_{n}^2b^2)/a^2\alpha_n^2$.

 The following theorem gives the recurrence relations for
$(\beta_{n})_{n\geq 0}$, $(\tilde{y}_n)_{n\geq 0}$, and $(\tilde{x}^2_n)_{n\geq 0}$.
\begin{theorem}
The sequence $(\beta_{n})_{n\geq 0}$ is a third-order  linear homogeneous  recurrence sequence
\begin{equation}\label{recbeta}
\beta_{n}= \left(2\lambda-1\right)\beta_{n-1}+ \left(2\lambda -1\right)\beta_{n-2}-\beta_{n-3} \quad (n\geq3).
\end{equation}
The sequence $(\tilde{y}_n)_{n\geq 0}$ satisfies the second-order linear homogeneous  recurrence sequence \eqref{eq:recurbi} and the sequence $(\tilde{x}^2_n)_{n\geq 0}$ is a third-order  linear homogeneous  recurrence sequence
\begin{equation}\label{xtild}
\tilde{x}^2_{n}=(4\lambda^2-1)\,\tilde{x}^2_{n-1}- (4\lambda^2-1)\, \tilde{x}^2_{n-2}+\tilde{x}^2_{n-3} \qquad (n\ge3).
\end{equation}
 The initial values are
$\beta_0$,  $\beta_1=2a\mu-\beta_0$, $\beta_{2}=8at^2\mu+\beta_{0}$,
$\tilde{y}_0=0$, 
$\tilde{x}^2_0=a^2$,   
\begin{align*} 
\tilde{y}_{1}&=\frac{\alpha_1(2+2a^2)}{a\beta_1^2+\alpha_1^2b^2},\\
\tilde{x}^2_1 &= \frac{\alpha_{1}^8a^2b^2(2a^2+2)^2}{(a^2\beta_{1}^2+\alpha_{1}^2b^2)^2(a\beta_{1}^2+\alpha_{1}^2b^2)^2}+a^2, \text{and}\\ 
\tilde{x}_{2}^2 &=  \frac{\alpha_2^2 b^2(a^2+\alpha_2^2)(-a^2b^2+\alpha_2^2+(b^2-\beta_2^2)x_2^2)}{(a^2\beta_2^2+\alpha_2^2b^2)^2}+a.
\end{align*}

\end{theorem}
\begin{proof}
We have $y_n$ is defined as follows: $y_{n}=\lambda \,y_{n-1}+2\mu \,r_{n-1}$ where $r_{n}$ is the radius of the circles. Using the tangency condition \eqref{eq:ellyn-yn} we get
\begin{equation}\label{beta}
\beta_{n}=2t^2 \,y_{n-1}+2\mu \,r_{n-1}-\beta_{n-1}.
\end{equation} 
After calculations, we obtain $\beta_n = 4t^2\,y_{n-1}+\beta_{n-2}$. Subtracting $\beta_{n-1}$ from $\beta_n$ we obtain \hspace{0.2cm}$\beta_n -\beta_{n-1}= 4t^2(y_{n-1}-y_{n-2})+\beta_{n-2}-\beta_{n-3}$.
Hence from Eq.~\eqref{eq:ellyn-yn}, we find \eqref{recbeta}.
The initial values come from \eqref{beta}.
The proofs of the second-order and third-order linear homogeneous  recurrence sequences of $(\tilde{y}_n)_{n\geq 0}$ and $(\tilde{x}^2_n)_{n\geq 0}$ are similar to the proof of Theorem \ref{th:re_eq_01}. Now $C={\alpha_{n}^4a^2b^2}/{(a^2\beta_{n}^2+\alpha_{n}^2b^2)^2}$.
\end{proof}
Equations \eqref{e:xk} and \eqref{xtild} are of the form $V_n=(\theta-1)V_{n-1}-(\theta-1)V_{n-2}+V_{n-3} \quad (n\geq3)$, and thus $W_n=V_n-V_{n-1}$ is a second-order linear recurrence $W_n=(\theta-2)W_{n-1}-W_{n-2}$ with $\theta=4\lambda^2$. We deduce an explicit form for $(V_n)_{n\geq 0}$ using $(W_n)_{n\geq 0}$ in the following theorem.
\begin{theorem} For all $n\geq 2$,
$V_n=V_0+\sum_{k=1}^nW_k$ .
\end{theorem}
\begin{proof}
The proof is left to the reader; 
it can be shown with some calculations.
The explicit terms of the $W_k$ are well known.
\end{proof}

\section{Associated integer sequences of chains}
In this section, we determine conditions to relate the circle and ellipse chains with integer sequences. In what follow, let $\beta_{0}=b$.

\subsection{Rectangular hyperbolas} \label{subs:reg}

\begin{corollary}\label{cor:rec_hyp}
	If  $\mathcal{H}$ is a rectangular hyperbola ($a=b$), then  $r_n=\alpha_n=\beta_n$,  $\hat{y}_n=\tilde{y}_n$, and $\hat{x}_n=\tilde{x}_n$ for all non-negative integers $n$. Moreover, the  recurrence sequence of $(r_n)_{n\geq 0}$,  $(y_n)_{n\geq 0}$, $(\hat{y}_n)_{n\geq 0}$,   and $(\hat{x}^2_n)_{n\geq 0}$ are, respectively, $\ell_n=6\ell_{n-1}-\ell_{n-2}\quad (n\geq2)$, with initial values $r_0=a$, $r_1=3a$, $y_0=0$, $y_1=4a$, $\hat{y}_0=0$, and $\hat{y}_1=2a$ and $\hat{x}_n^2=35\hat{x}_{n-1}^2-35\hat{x}_{n-2}+\hat{x}^2_{n-3} \quad (n\geq3)$,
with initial values  $\hat{x}^2_0=a^2$, $\hat{x}^2_1=5a^2$, and $\hat{x}_2^2=145a^2$.
\end{corollary}

For rectangular hyperbolas $\mathcal{H}$, from Corollary~\ref{cor:rec_hyp}, we deduce the following result:
\begin{theorem}\label{th:rech}
	If  $\mathcal{H}$ is a rectangular hyperbola, then   $(r_n)_{n\geq 0}$,  $(y_n)_{n\geq 0}$, $(\hat{y}_n)_{n\geq 0}$, and $(\hat{x}^2_n)_{n\geq 0}$ are integer sequences, respectively, if and only if 
	$a$ is a positive integer, $a=k/4$, $a=k/2$, and $a^2=k$, respectively, where $k$ is any positive integer. 
\end{theorem}

Now we give some examples of integer sequences. Some of them appear in
the OEIS \cite{OEIS} for $a=1$:  
$(r_n)_{n\geq0} =$ \seqnum{A001541}, 
$(y_n)_{n\geq0} =$ \seqnum{A005319}, 
$(\hat{y}_n)_{n\geq0} =$ \seqnum{A001542}, and 
 $(\hat{x}_n^2)_{n\geq0} = \{0,\text{\seqnum{A076218}}\}$ and 
for general $a$ with the conditions of Theorem~\ref{th:rech}, 
$(r_n)_{n\geq0} =  a\cdot\text{\seqnum{A001541}}$,
$(y_n)_{n\geq0} = a\cdot\text{\seqnum{A005319}}= 2a\cdot\text{\seqnum{A001542}}$, 
$(\hat{y}_n)_{n\geq0}= a\cdot\text{\seqnum{A001542}}$, and  
$(\hat{x}_n^2)_{n\geq0} = a^2\cdot \{0, \text{\seqnum{A076218}}\}$. For more sequences, see Table~\ref{tab:regular_hyp}.

We mention that among the sequences in Table~\ref{tab:regular_hyp} and Table~\ref{tab:regular_hyp2}, there are some sequences, e.g., \seqnum{A098706}, which are defined without any combinatorial or geometrical interpretation.
We provide a geometric interpretation.  We also note that $k\cdot(y_n)=2k\cdot(\hat{y}_n)$.
	
\begin{remark}
The sequence (\seqnum{A001109}), e.g., appears, as $(\hat{y}_n)_{n\geq0}$ and $(y_n)_{n\geq0}$ when $a=1/2$ and $a=1/4$, dealing with balancing numbers \cite{szal}.    
\end{remark}

\begin{table}[H] 
	\centering
	\begin{tabular}{|c||c|c|c|c|}
	\hline
	$a=b$ & $(r_n)_{n\geq0}$ & $(y_n)_{n\geq0}$ & $(\hat{y}_n)_{n\geq0}$ & $(\hat{x}_n^2)_{n\geq0}$\\
	\hline	\hline
	$1$ &\seqnum{A001541} & \seqnum{A005319}& \seqnum{A001542}& \seqnum{A076218}\\ 	\hline
	$a$ &$a\cdot$\seqnum{A001541} &$a\cdot$\seqnum{A005319}&$a\cdot$\seqnum{A001542}& $a^2\cdot$\seqnum{A076218}\\ 	\hline
	$2$ &\seqnum{A003499} & \seqnum{A081554}& \seqnum{A005319}& {\small  $\{4,20,580, 19604,\ldots\}$ }\\ 	\hline
	$3$ &\seqnum{A106329} &{\small  $\{0,12,72,420,\ldots\}$} & \seqnum{A075848}& {\small $\{9,45,1305,44109,\ldots\}$} \\ 	\hline
	$4$ &{\small  $\{4,12,68,396,\ldots\}$ }& $\{0,16,96,560,\ldots\}$ & \seqnum{A081554}& {\small $\{16,80,2320,78416,\ldots\}$}  \\ 		\hline
	$\sqrt{2}$ &-- & --& --& {\small \seqnum{A098706}$\setminus\{0\}$}\\ 		\hline		
\end{tabular}	
\caption{Integer sequences connected to the regular hyperbola.}
\label{tab:regular_hyp}
\end{table}



\subsection{Integer sequences associated with circle chains}

In order to generate only integer sequences, we state the following theorem.  
\begin{theorem}
If 	$t=(\sqrt{k-2})/2$, $k\geq3$, sequence $(r_n)_{n\geq 0}$, $a\in\mathbb{N}^+$ and $a\cdot k$ is even; $(y_n)_{n\geq 0}$, $a(k+2)$ is even; $(\hat{y}_n)_{n\geq 0}$,  $2a$ is a positive integer, then $b=2a/\sqrt{k-2}$ and the sequences $(r_n)_{n\geq 0}$, $(y_n)_{n\geq 0}$,  and $(\hat{y}_n)_{n\geq 0}$ are integer sequences.	
\end{theorem}

\begin{proof}
For $(r_n)_{n\geq0}$, we have $r_0=a$, so $a\in\mathbb{N}^+$ and $r_1=a(2t^2+1)$ is integer if and only if $(2t^2+1)= m/a$, where $m$ is a suitable positive integer. From this, we find $t^2=(m/a-1)/2$. For the first coefficient of \eqref{eq:recurbi}, we have $(4t^2+2)=2m/a=k$ where $k$ is a suitable positive integer. Now $m=(ak)/2$ implies that $ak$ is even and $t^2=\left(k-2\right)/4$. Moreover,
we have $t=\sqrt{k-2}/2$, where $t$ is positive if and only if  $k\geq3$. Obviously, $b=a/t$ comes from the definition of $t$.  

In the case where $(y_n)_{n\geq0}$, let $m=y_1=2a(t^2+1)$ be an integer. Then
$t^2=m/(2a)-1$ and $t=\sqrt{m/(2a)-1}$, where $m>2a$ and $a$ is a positive
real number. From the coefficient $(4t^2+2)=k$ with a suitable positive
integer, we have $m=a(k+2)/2$.   This implies that $a(k+2)$ is even and
$t^2=(k-2)/4$, $k\geq3$. Moreover, $t=\sqrt{k-2}/2$ and $b=a/t$.

For $(\hat{y}_n)_{n\geq0}$, $\hat{y}_1=2a=m$ and $(4t^2+2)=k$, we obtain that $2a$ is even and  \hbox{$t=\sqrt{k-2}/2$, $k\geq3$}.
\end{proof}

\begin{theorem}
	If $a^2$ is an integer and $t=k/2$, $k\geq1$, then the sequence $(\hat{x}_n^2)_{n\geq 0}$ consists of integers and  $b=a/t$.	
\end{theorem}
\begin{proof}
All the coefficients and initial values of \eqref{e:xk} are integers.
\end{proof}

 Table~\ref{tab:regular_hyp} and Table~\ref{tab:regular_hyp2} ($t=1$),  moreover, Table~\ref{tab:circle}, Table~\ref{tab:circle2}, and Table~\ref{tab:circle3} contain examples of integer sequences.   
 For $a=1$ and $b=2$, then $(y_n-r_n)_{n\geq1}$ is the bisection of Lucas sequence \seqnum{A002878}.
 
\begin{table}[H] 
	\centering
	\begin{tabular}{|c|c|c||c|}
	\hline
	$a$ & $b$ & t&$(\hat{x}_n^2)_{n\geq0}$\\
	\hline	\hline
	$\sqrt{2}$ & $1$ & $\sqrt{2}$  & {\small $\{2,18,1602,\ldots\}$} \\ 	
	\hline 
	$\sqrt{2}$ & $2$ & $\nicefrac{\sqrt{2}}{2}$   & {\small $\{2, 6, 66, 902,\ldots\}$} \\ 	
	\hline 
	$\sqrt{2}$ & $\nicefrac{\sqrt{2}}{2}$ & 2  & {\small $\{2, 34, 10370,\ldots\}$} \\ 
	\hline 
	
	$\sqrt{2}$ & $2\sqrt{2}$ &$\nicefrac{1}{2}$  & {\small $\{2, 4, 20, 130,\ldots\}$} \\ 	
	\hline   
\end{tabular}	
	\caption{Integer sequences associated with circle chains.}
	\label{tab:circle}
\end{table}

\begin{center}
	\resizebox{\columnwidth}{!}{
		\setlength{\tabcolsep}{1pt} 
	\begin{tabular}{|c|c|c||c|c|c|c|}
		\hline
$a$ & $b$ & t&$(r_n)_{n\geq0}$ & $(y_n)_{n\geq0}$ & $(\hat{y}_n)_{n\geq0}$ & $(\hat{x}_n^2)_{n\geq0}$\\
		\hline	\hline
$1$ & $2$ & $\nicefrac12$   & --- & --- & {\small $\{0,\text{\seqnum{A025169}}\}$;\seqnum{A111282},$n\geq1$}& \seqnum{A064170}  \\ 	\hline 		
$1$ & $\nicefrac12$ & $2$   &\seqnum{A023039} & {\small $\{0,10,180,\ldots\}$}& \seqnum{A207832}& {\small $\{1,17,5185,\ldots\}$} \\ 	\hline 
$1$ & $\nicefrac13$ & $3$  &\seqnum{A078986} & {\small $\{0,20,760,\ldots\}$}& {\small $\{0,2,76,2886,\ldots\}$} & {\small $\{1,37,51985,\ldots\}$} \\ 	\hline
$1$ &$\nicefrac23$  & $\nicefrac32$   & -- & -- & {\small $\{0,2,22,240\ldots\}$} & {\small $\{1,10,1090,\ldots\}$} \\ 	\hline
$1$ & $\nicefrac14$ &$4$ &\seqnum{A099370} & {\small $\{0,34,2244,\ldots\}$}& {\small $\{0,2,132,8710,\ldots\}$} & {\small $\{1,65,278785,\ldots\}$} \\ 	\hline

$2$ & $1$ &2 &\seqnum{A087215} & \seqnum{A004292}, $n\geq1$& \seqnum{A060645}& {\small $\{4,68,20740,\ldots\}$} \\ 	\hline 

$2$ & $4$ &  $\nicefrac12$  &\seqnum{A005248} & \seqnum{A201157}& {\small $\{0,4,12,32,\ldots\}$}& {\small $\{4,8,40,260,\ldots\}$} \\ 	\hline 

$2$ & $\nicefrac12$ &4 & {\small $\{2,66,4354,\ldots\}$} & \seqnum{A004298}, $n\geq1$ & {\small $\{0,4,264,17420,\ldots\}$}& {\small $\{4,260,1115140,\ldots\}$} \\ 	\hline 

$2$ & $\nicefrac13$ &6 & {\small$\{2,146,21314,\ldots\}$} & {\small $\{0,148,21608,\ldots\}$}& {\small $\{0,4,584,85260,\ldots\}$}& {\small $\{4,580,12278020,\ldots\}$} \\ 	\hline 

$2$ & $\nicefrac23$  & $3$  & {$\{2,\seqnum{A239364}\}$} & {\small $\{0,13,143,\ldots\}$}& {\small $\{0,4,44,480,\ldots\}$}& {\small $\{4,148,207940,\ldots\}$} \\ 	\hline 

$2$ & $\nicefrac43$  & $\nicefrac32$  & \seqnum{A057076} & {\small $\{0,40,1520,\ldots\}$}& {\small $\{0,4,152,5772,\ldots\}$}& {\small $\{4,40,4360,\ldots\}$ } \\ 	\hline

$3$ & $1$  &3& {\small$\{3,57,2163,\ldots\}$} & {\small $\{0,60,2280,\ldots\}$}& \seqnum{A084070} & {\small $\{9,333,467865,\ldots\}$} \\ 	\hline 

$3$ & $6$  & $\nicefrac12$  & --- & --- & \seqnum{A099857}, $n\geq1$ & {\small $\{9, 18, 90, 585,\ldots\}$} \\ 	\hline 

$4$ & $1$ &4 & {\small  $\{4, 132, 8708,\ldots\}$} & {\small $\{ 0, 136, 8976,\ldots\}$}& {\small $\{0, 8, 528,34840,\ldots\}$} & {\small $\{16, 1040, 4460560,\ldots\}$} \\ 	\hline

$4$ & $2$ &2 & {\small$\{4,36,644,\ldots\}$} & {\small $\{0, 40, 720,\ldots\}$}& \seqnum{A134492} & {\small $\{16,272,82960,\ldots\}$} \\ 	\hline 

$4$ & $8$  & $\nicefrac12$  & {\small  $\{4, 6, 14, 36,\ldots\}$} & {\small $\{0, 10, 30, 80,\ldots\}$}& {\small $\{0, 8, 24, 64,\ldots\}$} & {\small $\{16, 32, 160,\ldots\}$} \\ 	\hline
$1$ & $\sqrt{2}$ & $\nicefrac{\sqrt{2}}{2}$ & \seqnum{A001075} & \seqnum{A005320}& \seqnum{A052530}& \seqnum{A011922} \\ 	\hline
$1$ & $\nicefrac{\sqrt{2}}{2}$  & $\sqrt{2}$ & \seqnum{A001079} & \seqnum{A122653}& \seqnum{A001078}& {\small $\{1, 9, 801,78409,\ldots\}$} \\ 	\hline
$2$ & $\sqrt{2}$ & $\sqrt{2}$  & \seqnum{A087799} & \seqnum{A004291}, $ n\geq1$  & \seqnum{A122652} & {\small $\{4, 36, 3204,\ldots\}$} \\ 	\hline 

$2$ & $\nicefrac{\sqrt{2}}{2}$ & $2\sqrt{2}$  &{\small $\{2,34,1154,\ldots\}$} & \seqnum{A004294}, $ n\geq1$ & \seqnum{A202299} & {\small $\{4, 132, 147972, \ldots\}$} \\ 	\hline

$2$ & $2\sqrt{2}$& $\nicefrac{\sqrt{2}}{2}$   & \seqnum{A003500} & \seqnum{A001352}, $ n\geq1$ & \seqnum{A231896}  & {\small $\{4, 12, 132,\ldots\}$} \\ 	\hline 
	\end{tabular}
}
\captionof{table}{Integer sequences associated with circle chains.}
	\label{tab:circle2}
\end{center}


\begin{figure}[H]
\begin{minipage}{.53\linewidth}
	\begin{tabular}{|c|c||c|c|c|}
	\hline
$a$ & $b$ & t & $(y_n)_{n\geq0}$ & $(\hat{y}_n)_{n\geq0}$ \\
		\hline	\hline

$\nicefrac12$ & $1$  & $\nicefrac12$& --- & \seqnum{A001906} \\
\hline

$\nicefrac12$ & $\nicefrac13$  & $\nicefrac32$ & --- & {\small $\{0,\text{\seqnum{A004190}}\}$} \\
 	\hline

$\nicefrac12$ & $\nicefrac14$&2 & {\small $\{0,5,90,\ldots\}$}  & \seqnum{A049660}\\
\hline

$\nicefrac12$ & $\nicefrac15$  & $\nicefrac52$  &--- & {\small $\{0\text{,\seqnum{A049660}}\}$} \\
	\hline
$\nicefrac12$ & $\nicefrac16$&3 &{\small $\{0,10,380,\ldots\}$}  & {\small $\{0,\text{\seqnum{A078987}}\}$} \\	
\hline
$\nicefrac12$ & $\nicefrac17$  & $\nicefrac72$  & ---  & \seqnum{A097836}\\
\hline
$\nicefrac12$ & $\nicefrac18$  & $4$ &{\small $\{0,17,1122,\ldots\}$}  & {\small $\{0,\text{\seqnum{A097316}}\}$} \\
 	\hline
$\nicefrac12$ & $\nicefrac19$ &  $\nicefrac92$ & ---  & {\small $\{0,\text{\seqnum{A097839}}\}$}\\
\hline
$\nicefrac12$ & $\nicefrac{1}{10}$  & 5 &{\small $\{0,26,2652,\ldots\}$}  & {\small $\{0,\text{\seqnum{A097725}}\}$}\\
\hline
$\nicefrac12$ & $\nicefrac{1}{11}$  & $\nicefrac{11}{2}$ & --- & {\small $\{0,\text{\seqnum{A049670}}\}$} \\
 	\hline
$\nicefrac12$ & $\nicefrac{1}{13}$& $\nicefrac{13}{2}$  & --- & {\small $\{0,\text{\seqnum{A097844}}\}$} \\
	\hline
\end{tabular}
\captionof{table}{Integer sequences associated with circle chains.}
\label{tab:circle3}
\end{minipage}
\hfill 
\begin{minipage}[]{.4\linewidth}
\resizebox{\columnwidth}{!}{
		\setlength{\tabcolsep}{1pt}
	\begin{tabular}{|c|c|c|}
	\hline
$a=b$ & $(y_n)_{n\geq0}$ & $(\hat{y}_n)_{n\geq0}$\\
		\hline	\hline

$\nicefrac12$ & \seqnum{A001542}& \seqnum{A001109}\\
 \hline		
		$\nicefrac32$ & \seqnum{A075848}& \seqnum{A106328}\\
		\hline	
		$\nicefrac52$& {\small \{$0,10,60,350,\ldots\}$} & \seqnum{A276598}\\
		\hline			
		$\nicefrac72$ &{\small $\{0,\seqnum{A273182}\}$} &   \seqnum{A054890},{\small$n\geq1$} \\	
		\hline
		$\nicefrac92$&  {\small $\{0,18,108,630,\ldots\}$} & \seqnum{A276602}\\
				\hline		
		$\nicefrac14$ & \seqnum{A001109} & --\\		\hline
		$\nicefrac34$& \seqnum{A106328}& --\\		\hline	
		$\nicefrac54$ & \seqnum{A276598}& --\\		\hline
		$\nicefrac74$ & \seqnum{A054890}, $n\geq1$& --\\		
  		\hline
	
	\end{tabular}}
\captionof{table}{Integer sequences connected to the regular hyperbola.}
\label{tab:regular_hyp2}
\end{minipage}
\end{figure}

\subsection{Integer sequences associated with ellipse chains}

\begin{theorem} 
For positive integers $a,b$, if $b$ divides $a$ then the sequence $(\beta_{n})_{n\geq0}$ is an integer sequence and its recurrence is
\begin{equation}\label{receq}
\begin{array}{ll}
\beta_{n}=(4t^2+1)\beta_{n-1}+(4t^2+1)\beta_{n-2}-\beta_{n-3} \quad (n\geq3),
\end{array}
\end{equation}
and the initial values are 
$\beta_0=b$, $\beta_1=2a\mu-b$, and $\beta_2= 8at^2\mu+b$.
\end{theorem}
\begin{proof}
We notice that the initial values of the sequence are integers. Thus, the  coefficients of recurrence relation \eqref{receq} are also integers, 
which  guarantees that all the other terms of the sequence are integers.
\end{proof}
Equations \eqref{recbeta}
 and \eqref{receq} are of the form $V_n=(\theta-1)V_{n-1}+ (\theta-1)V_{n-2}-V_{n-3}, \quad (n\geq3)$, and thus $W_n = V_n+V_{n-1}$ is a second-order linear recurrence $W_n=\theta W_{n-1}-W_{n-2}$. Here $\theta=2\lambda$ and $4t^2$,
 respectively.
We deduce an explicit form for $(V_n)_{n\geq0}$ using $(W_n)_{n\geq0}$ and 
leaving the proof to the reader in the following theorem:
\begin{theorem} For all $n \geq 2$, we have
$V_n=(-1)^nV_0+ \sum_{k=1}^n(-1)^{n-k}W_k$.
\end{theorem} 
The explicit terms of $W_k$ are well known.  

We give some integer recurrence sequences for $(\beta_{n})_{n \geq 0}$ in Table~\ref{tab:ellipse}. If $t=1$, so $a=b$, then our hyperbola is a rectangular hyperbola, and it holds for the integer sequences associated with ellipse chain not only for the sequence $(\beta_{n})_{n \geq 0}$, but also for the sequences $\alpha_n$, $\beta_n$, $\tilde{y}_n$, and $\tilde{x}_n$. See Subsection~\ref{subs:reg}.
\begin{center}
		\begin{tabular}{|c|c|c||c|}
			\hline
			$a$ & $b$& $t$& $(\beta_{n})_{n\geq0}$ \\	\hline	\hline
			$a$ & $a$& $1$& in Table \ref{tab:regular_hyp} \\	\hline	
			$2$ &$1$ & $2$&$\{1,19,321,5779,103681,1860499,\ldots\}$\\ 	\hline
			$3$ &$1$ & $3$&$\{1,59,2161,82139,3119041,118441499,\ldots\}$ \\ 	\hline
			$4$ &$1$ & $4$&$\{1,135,8705,574599,37914625,2501790855,\ldots\}$ \\ 	\hline
			$4$ &$2$ & $2$&$\{2,38,642,11558,207362,3720998,\ldots\}$  \\ 		\hline
		\end{tabular}
	\captionof{table}{Integer sequences associated with ellipse chains.} \label{tab:ellipse}
\end{center}

\section{Acknowledgments}

For H. B. and S. M. T., {this work was supported by the grand of DGRSDT, number C0656701}. 
For L. N. {this work has been performed under the auspices of the program ``{\sc Efop}-3.6.1-16-2016-00018---Improving the role of the research $+$ development $+$ innovation in the higher education through institutional developments assisting intelligent specialization in Sopron and Szombathely''.}


\begin{thebibliography}{9}
\bibitem{szal} M. Alp, N. Irmak, and L. Szalay, Balancing diophantine triples with distance 1, {\it Period. Math. Hung.} {\bf 71} (2015), 1--10. 
\bibitem{BNT} H. Belbachir, L. N\'emeth, and  S. M. Tebtoub, Integer sequences and ellipse chains inside a hyperbola, {\it Ann. Math. Inform.} {\bf 52} (2020), \url{https://doi.org/10.33039/ami.2020.06.002}.	
\bibitem{Lucca} G. Lucca, Integer sequences and circle chains inside a hyperbola, {\it Forum Geom.} {\bf 19} (2019),  11--16.
\bibitem{OEIS} N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, \url{https://oeis.org}.
\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B37; Secondary 52C26.

\noindent \emph{Keywords: }  circle chain, ellipse chain, ellipsoid chain, recurrence relation, integer sequence.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A001075},
\seqnum{A001078},
\seqnum{A001079},
\seqnum{A001109},
\seqnum{A001352},
\seqnum{A001541},
\seqnum{A001542},
\seqnum{A001906},
\seqnum{A002878},
\seqnum{A003499},
\seqnum{A003500},
\seqnum{A004190},
\seqnum{A004291},
\seqnum{A004292},
\seqnum{A004294},
\seqnum{A004298},
\seqnum{A005248},
\seqnum{A005319},
\seqnum{A005320},
\seqnum{A011922},
\seqnum{A023039},
\seqnum{A025169},
\seqnum{A049660},
\seqnum{A049670},
\seqnum{A052530},
\seqnum{A054890},
\seqnum{A057076},
\seqnum{A060645},
\seqnum{A064170},
\seqnum{A075848},
\seqnum{A076218},
\seqnum{A078986},
\seqnum{A078987},
\seqnum{A081554},
\seqnum{A084070},
\seqnum{A087215},
\seqnum{A087799},
\seqnum{A097316},
\seqnum{A097725},
\seqnum{A097836},
\seqnum{A097839},
\seqnum{A097844},
\seqnum{A098706},
\seqnum{A099370},
\seqnum{A099857},
\seqnum{A106328},
\seqnum{A106329},
\seqnum{A111282},
\seqnum{A122652},
\seqnum{A122653},
\seqnum{A134492},
\seqnum{A201157},
\seqnum{A202299},
\seqnum{A207832},
\seqnum{A231896},
\seqnum{A239364},
\seqnum{A273182},
\seqnum{A276598}, and
\seqnum{A276602}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received April 8 2020;
revised versions received  August 27 2020; September 10 2020.
Published in {\it Journal of Integer Sequences}, September 11 2020.

\bigskip
\hrule
\bigskip

\noindent
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\end{document}



