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\begin{center}
	\vskip 1cm{\Large\bf Total Positivity of Toeplitz Matrices Involving\\
	\vskip .1in
	Generalized Hyper-Fibonacci Numbers
	}
	\vskip 1cm
		Abdelhak Taane \\
	University of Kasdi Merbah \\ 
	Department of Mathematics\\
	Laboratory of Applied Mathematics\\
	Ouargla, 30000\\ Algeria\\
	\href{mailto:abdelhak.taane@gmail.com}{\tt abdelhak.taane@gmail.com}\\		
	\ \\
	Ihab-Eddine Djellas \\
	CERIST Scientific and Technical Information Research Center\\
	Algiers\\
	 Algeria \\
	 and \\
	University of Sciences and Technology Houari Boumediene (USTHB)\\ 				Faculty of Mathematics \\
	RECITS Laboratory\\
	P. O. Box 32, El Alia, 16111\\
	Bab Ezzouar, Algiers\\
	Algeria\\
	\href{mailto:ihabeddinedjellas@gmail.com}{\tt ihabeddinedjellas@gmail.com}\\
	\ \\
	Mohammed Mekkaoui\\ 
	{\'E}cole Normale Sup{\'e}rieure \\
	Department of Mathematics\\ 
    EDPNLHM Laboratory\\
	 B. P. 92, 	Vieux Kouba, 16050\\
	Kouba, Algiers\\ 
	Algeria\\
	\href{mailto:mohammed.mekkaoui@g.ens-kouba.dz}{\tt mohammed.mekkaoui@g.ens-kouba.dz} \\
	\href{mailto:mohammed91mekkaoui@gmail.com}{\tt mohammed91mekkaoui@gmail.com} \\
\end{center}

\vskip .2 in
\begin{abstract}
We study the total positivity of Toeplitz matrices built from generalized hyper-Fibonacci sequences of a fixed generation. Using this approach, we prove that each sequence becomes log-concave beyond a certain point for every generation. We also present several notable special cases that illustrate the scope of our results.
\end{abstract}
\section{Introduction}
\label{Introduction}
Let $M$ be an infinite real matrix. If all minors of $M$ have positive 
determinants, we say that $M$ is \textit{totally positive}, or simply $\text{TP}$. If all minors of $M$ of every order $\leq r$ for some integer $r$ are positive, we say that $M$ is \textit{totally positive of order $r$}, or simply $\text{TP}_r$. 

Several studies in the literature have addressed the total positivity of matrices associated with various classes of recurrent sequences. In particular, many combinatorial triangular matrices have been shown to possess the total positivity property. For example, Wang and Wang \cite{Catalan} investigated the total positivity of the Catalan triangle. Additionally, Ahmia and Belbachir \cite{strongq-log-convexity} proved that the generalized Pascal triangle is \(\text{TP}_2\). For additional results concerning triangular matrices and Riordan arrays, we refer the reader to~\cite{Combinatoricsandtotalpositivity,Totalpositivityofrecursivematrices,TotalpositivityofRiordanarrays}.

A \textit{Toeplitz} matrix $T=[t_{i,j}]$ is a (finite or infinite) matrix whose entries satisfy $t_{i+1,j+1}=t_{i,j}$. In the finite case, that is
\begin{equation*}
	T =
\begin{pmatrix}
		t_{0} & t_{1} & \cdots & t_{n}\\
		t_{-1} & t_{0} & \cdots &t_{n-1}\\
		\vdots & \vdots & \ddots & \vdots\\
		t_{-n} & t_{-n+1} & \cdots & t_{0}\\
\end{pmatrix}.
\end{equation*}
A \textit{Hankel} matrix $H=[h_{i,j}]$ is a (finite or infinite) matrix whose entries satisfy $t_{i+1,j-1}=t_{i,j}$. In the finite case, that is
\begin{equation*}	
H =
\begin{pmatrix}
		h_{0} & h_{1} & \cdots & h_{n}\\
		h_{1} & h_{2} & \cdots &h_{n+1}\\
		\vdots & \vdots & \ddots & \vdots\\
		h_{n} & h_{n+1} & \cdots & h_{2n}\\
\end{pmatrix}.
\end{equation*}


Recall that a sequence $(b_n)_{n\geq 0}$ of positive numbers is \textit{log-concave} (resp., \textit{log-convex}) if $b_{i+1}^2\geq b_{i}b_{i+2}$ for all $i\geq 0$ (resp., \ $b_{i+1}^2\leq b_{i}b_{i+2}$ for all $i\geq 0$).
The $\text{TP}_2$ property of Hankel and Toeplitz matrices is closely connected to the log-convexity and log-concavity, respectively, of the associated sequences.
The properties of log-concavity and log-convexity have been widely studied in the literature due to their importance in combinatorics. Recent advances have introduced novel techniques for establishing such properties; see, for instance, \cite{Doslis,Doslis2,Zheng1,Catalan}. These expansions complement the classical results given by Stanley  \cite{Stanly}.

The total positivity of Toeplitz and Hankel matrices is among the important properties of these matrices; such properties can find applications in a number of fields, including chemistry, electrical networks, game theory, differential equations, stochastic processes, orthogonal polynomials, combinatorics, quantum groups, algebraic geometry, symmetric functions, and representation theory; see \cite{Combinatoricsandtotalpositivity,Fallat,Gasca,P2025,Pinkus} for some applications.

Let $p\geq 1$, $a\geq 0$, and $b\geq 0$ be  integers. We consider the generalized  Fibonacci sequence $(W_{p,n})_{n\geq 0}$ defined recursively as follows:
\begin{align}W_{p,n}=pW_{p,n-1}+W_{p,n-2}\; (n\geq 2),\quad   W_{p,0}=a,\; W_{p,1}=b.\label{p-Horadam} 
\end{align}

\begin{remark} Some particular cases of the above sequence are
	\begin{enumerate}
		\item For $(a,b)=(0,1)$, we obtain the $p$-Fibonacci numbers $F_{p,n}$ defined by Falc\'on \cite{Falcan}.
		\item For  $(a,b)=(2,p)$, we obtain the $p$-Lucas numbers $L_{p,n}$ defined by  Falc\'on and Plaza \cite{Falcan-Lucas}.
	\end{enumerate}
Furthermore, for $p=1$, the sequences  $(F_{1,n})_{n\geq 0}=(F_n)_{n\geq 0}$ and $(L_{1,n})_{n\geq 0}=(L_{n})_{n\geq 0}$ correspond to the classical Fibonacci and Lucas sequences, respectively.		
We refer to \cite{Falcan-Lucas,Falcan,horadam1965basic,Mekkaoui,Taane,Vajda} for some properties and identities involving the $p$-Fibonacci and $p$-Lucas numbers.
\end{remark}
Let $r\geq 0$ be a fixed integer. Dil and  Mez\H{o} \cite{Dil} introduced the hyper-Fibonacci numbers $F_n^{(r)}$ and the hyper-Lucas numbers $L_n^{(r)}$ as follows:
\begin{align*}
	F_n^{(r)}&=\sum_{k=0}^{n}F_k^{(r-1)}, \; F_0^{(r)}=0, \; F_1^{(r)}=1,\;F_n^{(0)}=F_n,\\	L_n^{(r)}&=\sum_{k=0}^{n}L_k^{(r-1)}, \; L_0^{(r)}=2,
	\; L_1^{(r)}=2r+1,
	\;L_n^{(0)}=L_n.
\end{align*}

Let $p\geq 1$ and $r\geq 1$ be two integers. The generalized hyper-Fibonacci numbers $W_{p,n}^{(r)}$ associated with the generalized Fibonacci numbers
$W_{p,n}$ are defined as follows \cite{Ait1,bahcsi2014symmetric}:
\begin{align}\label{def_generalized hyperFib}
	W_{p,n}^{(r)}=\sum_{k=0}^{n}p^{n-k}W_{p,k}^{(r-1)},\quad W_{p,n}^{(0)}=W_{p,n},\quad  W_{p,0}^{(r)}=a, \quad W_{p,1}^{(r)}=par+b.
\end{align}
\begin{remark}
	Some particular cases of the generalized hyper-Fibonacci numbers are 
	\begin{enumerate}
		\item For $(a,b)=(0,1)$, we get the hyper $p$-Fibonacci numbers $F_{p,n}^{(r)}$; see \cite{Ait1,Ait3,Bivari}.
		\item For $(a,b)=(2,p)$, we get the hyper $p$-Lucas numbers $L_{p,n}^{(r)}$; see \cite{Ait2}.
	\end{enumerate}
Furthermore, for $p=1$, we have  $F_{1,n}^{(r)}=F_{n}^{(r)}$
		and  $L_{1,n}^{(r)}=L_{n}^{(r)}$. Also, for $p=2$, the numbers $F_{2,n}^{(r)}=P_{n}^{(r)}$
		and  $L_{2,n}^{(r)}=Q_{n}^{(r)}$ correspond to the $n^{\rm th}$ hyper-Pell and hyper-Pell-Lucas numbers, respectively; see \cite{AhmiaPell}.	
\end{remark}
Do\v{s}li\'c et  al.\ \cite{Martinjak} studied the total positivity of a class of Toeplitz matrices composed of the hyper-Fibonacci sequence $(F_{n}^{(r)})_{n\geq 0}$. A similar study but for the hyper-Lucas sequence $(L_{n}^{(r)})_{n\geq 0}$ has been studied by Rezig and Ahmia \cite{AhmiaRezig}. 
The aim of our work is to extend these results to the generalized hyper-Fibonacci numbers $(W_{p,n}^{(r)})_{n\geq 0}$. To do so, we define the following $m\times m$ Toeplitz matrix:
\begin{equation*}
	T_{m,n}^{(r)} :=
\begin{pmatrix}
		W_{p,n}^{(r)} & W_{p,n-1}^{(r)} & \cdots & W_{p,n-m+1}^{(r)}\\
		W_{p,n+1}^{(r)} & W_{p,n}^{(r)} & \cdots & W_{p,n-m+2}^{(r)}\\
		\vdots & \vdots & \ddots & \vdots\\
		W_{p,n+m-1}^{(r)} & W_{p,n+m-2}^{(r)} & \cdots & W_{p,n}^{(r)}\\
\end{pmatrix}.
\end{equation*}
In Section \ref{Preliminary results}, we present several identities and auxiliary results that we use in this work. In Section \ref{Sec3}, we establish the positivity of the determinant $\det\bigl(T_{m,n}^{(r)}\bigr)$ under the condition \(m \leq r+1\). This, in turn, implies the log-concavity of the sequences \((W_{p,n}^{(r)})_{n \geq 0}\) for all positive integers \(r\) and sufficiently large \(n\). Finally, in Section \ref{Sec4}, we examine the total positivity of the Toeplitz matrix \(T_{r+2,n}^{(r)}\).

Throughout this paper, we assume that  $p\geq1$, $a\geq 0$, and  $b> 0$  are integers.
\section{Preliminary results}\label{Preliminary results}
In this section, we present some of the results we need for the rest of the paper.

Initial minors play a fundamental role in the study of total positivity of matrices. Let $I$ and $J$ denote the row set and column set, respectively. A \textit{minor} $b_{I,J}$ is called \textit{initial} if both $I$ and $J$ consist of consecutive indices and  $I\cup J$ contain $1$. Consequently, each matrix entry is the lower-right corner of exactly one initial minor.
The result established by Gasca and Pe\~na \cite{initialminors} provides a useful method for testing the total positivity of matrices:
\begin{theorem}\cite{initialminors}
	A matrix is totally positive if and only if all its initial minors
	are positive.
\end{theorem}

Numerous identities involving generalized Fibonacci numbers have been established in the literature. Let $\Delta=a^2+pab-b^2$, $ \alpha=\frac{p+\sqrt{p^2+4}}{2}$, and $\beta=\frac{p-\sqrt{p^2+4}}{2}$. Then we have
\begin{lemma}
	Let $m\geq 0$ and $n\geq 0$ be two integers. Then the following identities hold.
	\begin{align}
		&W_{p,n}=A \alpha^{n}+B\beta^{n},\; \text{where}\; A=\frac{b-a\beta }{\sqrt{p^2+4}},\; B=\frac{a \alpha-b}{\sqrt{p^2+4}},\\
		&F_{p,m-2}W_{p,n-2}-F_{p,m-3}W_{p,n-1}=(-1)^{m-1}W_{n-m+1},\label{EQQ}\\
		&	W_{p,n}W_{p,n+2}-W_{n+1}^{2}=(-1)^{n}\Delta\label{cassini},\\
		&\lim_{n\to +\infty}\frac{W_{p,n+1}}{W_{p,n}}=\alpha.
	\end{align}
\end{lemma}


\begin{lemma}\cite{Ait1} Let $r\geq 1$ be an integer. The following identities hold
	\begin{align}
		W_{p,n}^{(r)}&=pW_{p,n-1}^{(r)}+W_{p,n}^{(r-1)}\label{Id2}\;\;(n\geq1),\\
		W_{p,n-r}^{(r)}&=W_{p,n+r}-S_r \;\;(n\geq r), \label{Id3}
	\end{align}
	where
	$S_r=p^{n}\sum_{k=0}^{r-1}\binom{n-k-1}{r-k-1}\frac{W_{p,2k+1}}{p^{r-1}}$.
\end{lemma}

\begin{remark}\label{rem1}
	In \eqref{Id3}, the numbers $S_r$ can be expressed as $S_r=p^{n}f_r(n)$, where $f_r(n)$ is a polynomial in $n$.
\end{remark}

Let $r\geq 0$ and $n\geq 0$ be two integers. We consider the following Hankel matrix of order $r+2$:
\begin{equation*}
H_{r,n} =
\begin{pmatrix}
		W_{p,n}^{(r)} & W_{p,n+1}^{(r)} & \cdots & W_{p,n+r+1}^{(r)}\\
		W_{p,n+1}^{(r)} & W_{p,n+2}^{(r)} & \cdots & W_{p,n+r+2}^{(r)}\\
		\vdots & \vdots & \ddots & \vdots\\
		W_{p,n+r+1}^{(r)} & W_{p,n+r+2}^{(r)} & \cdots & W_{p,n+2r+2}^{(r)}\\
\end{pmatrix}.
\end{equation*} 
\begin{proposition}\cite{Ait1}\label{propo2}
	Let $r\geq 0$ and $n\geq 0$ be integers. We have
	\begin{align}
		&\det(H_{r,n})=(-1)^{n+1+\lfloor\frac{r+3}{2} \rfloor}p^{nr+r^2}b^r\Delta.
	\end{align}
	
\end{proposition}

\section{Positivity of Toeplitz determinants}\label{Sec3}
In this section, we discuss the positivity of determinants of some classes of Toeplitz matrices where the entries are generalized hyper-Fibonacci numbers of the $r^{\rm th}$ generation. 

Let $T_{m,n}^{(r)}$ denote the Toeplitz matrix of order $m$ consisting of generalized hyper-Fibonacci numbers of the $r^{\rm th}$ generation
\begin{equation*}
T_{m,n}^{(r)} =
\begin{pmatrix}
	W_{p,n}^{(r)} & W_{p,n-1}^{(r)} & \cdots & W_{p,n-m+1}^{(r)}\\
	W_{p,n+1}^{(r)} & W_{p,n}^{(r)} & \cdots & W_{p,n+-m+2}^{(r)}\\
	\vdots & \vdots & \ddots & \vdots\\
	W_{p,n+m-1}^{(r)} & W_{p,n+m-2}^{(r)} & \cdots & W_{p,n}^{(r)}\\
\end{pmatrix},	
\end{equation*}
with the condition  $m\leq r+1$.
\begin{theorem}\label{lemma}
	Let $m\geq 1$ be an integer. There exists a positive integer $n_m$ such that $\det\bigl( T_{m,n}^{(m-1)}\bigr) >0$  for all $n\geq n_m$.
\end{theorem}
\begin{proof}
	For $m=2$, using \eqref{Id2}, we obtain
	\begin{align*}
		\det\bigl( T_{2,n}^{(1)}\bigr)  &:=
		\begin{vmatrix}
				W_{p,n}^{(1)} & W_{p,n-1}^{(1)}\\
				W_{p,n+1}^{(1)} & W_{p,n}^{(1)} 
		\end{vmatrix} 
		=\begin{vmatrix}
				W_{p,n} & W_{p,n-1}^{(1)}\\
				W_{p,n+1} & W_{p,n}^{(1)} 
		\end{vmatrix}=\begin{vmatrix}
				W_{p,n} & W_{p,n-1}^{(1)}\\
				W_{p,n-1} & W_{p,n}
		\end{vmatrix} .
	\end{align*}
	Thus, by using \eqref{Id3}, we have
	\begin{align*}
		\det\bigl( T_{2,n}^{(1)}\bigr) =
		\begin{vmatrix}
				W_{p,n} & W_{p,n+1}-S_1\\
				W_{p,n-1} & W_{p,n}
		\end{vmatrix}.
	\end{align*}
	According to \eqref{cassini}, the last determinant is positive starting from a certain value $n_2$. 
	
	In the rest of the proof, we assume that $m\geq3$.
	Using \eqref{Id2} and  some elementary transformations on columns, we get
	\begin{align*}
		\det\bigl( T_{m,n}^{(m-1)}\bigr)  
		&
		:=
		\begin{vmatrix}
				W_{p,n}^{(m-1)} & W_{p,n-1}^{(m-1)} & \cdots & W_{p,n-m+1}^{(m-1)}\\
				W_{p,n+1}^{(m-1)} & W_{p,n}^{(m-1)} & \cdots & W_{p,n-m+2}^{(m-1)}\\
				\vdots & \vdots & \ddots & \vdots\\
				W_{p,n+m-1}^{(m-1)} & W_{p,n+m-2}^{(m-1)} & \cdots & W_{p,n}^{(m-1)}\\
		\end{vmatrix}\\
		&=\begin{vmatrix}
				W_{p,n} & W_{p,n-1}^{(1)}&W_{p,n-2}^{(2)} & \cdots & W_{p,n-m+1}^{(m-1)}\\
				W_{p,n+1} & W_{p,n}^{(1)}&W_{p,n-1}^{(2)} & \cdots & W_{p,n-m+2}^{(m-1)}\\
				\vdots & \vdots &\vdots&  & \vdots\\
				W_{p,n+m-1} & W_{p,n+m-2}^{(1)} & W_{p,n+m-3}^{(2)}& \cdots & W_{p,n}^{(m-1)}\\
		\end{vmatrix} .
	\end{align*}

	Again, using \eqref{Id2} and  some elementary transformations on rows,   we get
	\begin{align*}
		\det\bigl( T_{m,n}^{(m-1)}\bigr)&=
		\begin{vmatrix}
				W_{p,n} & W_{p,n-1}^{(1)}&W_{p,n-2}^{(2)} & \cdots & W_{p,n-m+1}^{(m-1)}\\
				W_{p,n-1} & W_{p,n}&W_{p,n-1}^{(1)} & \cdots & W_{p,n-m+2}^{(m-2)}\\
				\vdots & \vdots &\vdots&  & \vdots\\
				W_{p,n-m+2} & W_{p,n-m+3} & W_{p,n-m+4}& \cdots & W^{(1)}_{n-1}\\
				W_{p,n-m+1} & W_{p,n-m+2} & W_{p,n-m+3}& \cdots & W_{p,n}
		\end{vmatrix}. 
	\end{align*}
	Now, using \eqref{Id3}, we write
	\begin{align*}
		\det\bigl( T_{m,n}^{(m-1)}\bigr) =
\begin{vmatrix}
				W_{p,n} & W_{p,n+1}-S_1&W_{p,n+2}-S_2 & \cdots & W_{p,n+m-1}-S_{m-1}\\
				W_{p,n-1} & W_{p,n}&W_{p,n+1}-S_1 & \cdots & W_{p,n+m-2}-S_{m-2}\\
				\vdots & \vdots &\vdots&  & \vdots\\
				W_{p,n-m+2} & W_{p,n+m+3} & W_{p,n-m+3}& \cdots & W_{p,n+1}-S_1\\
				W_{p,n-m+1} & W_{p,n-m+2} & W_{p,n-m+3}& \cdots & W_{p,n}
		\end{vmatrix} .
	\end{align*}
	Using the recurrence relation \eqref{p-Horadam} on columns, we obtain

\resizebox{.9\linewidth}{!}{
  \begin{minipage}{\linewidth}
  \begin{align*}
			&\det\bigl( T_{m,n}^{(m-1)}\bigr) =
		\begin{vmatrix}
					S_2-pS_1 & S_3-pS_2-S_1&  \cdots &W_{p,n+m-2}-S_{m-2} & W_{p,n+m-1}-S_{m-1}\\
					S_1 & S_2-pS_1& \cdots & W_{p,n+m-3}-S_{m-3}& W_{p,n+m-2}-S_{m-2}\\
					0 & S_1& \cdots & W_{p,n+m-4}-S_{m-4}& W_{p,n+m-3}-S_{m-3}\\
					\vdots & \vdots && \vdots & \vdots\\
					0&0 &\cdots& W_{p,n} & W_{p,n+1}-S_1\\
					0 &0 &  \cdots &W_{p,n-1}& W_{p,n}
			\end{vmatrix}.
\end{align*}
  \end{minipage}
}

Again, using \eqref{p-Horadam} on rows, we obtain

\resizebox{.95\linewidth}{!}{
  \begin{minipage}{\linewidth}
  \begin{align*}
		\det\bigl( T_{m,n}^{(m-1)}\bigr)	&=\begin{vmatrix}
					S_2-2pS_1 & S_3-2pS_2+(p^2-2)S_1&  \cdots &-S_{m-2}+pS_{m-3}+S_{m-4}& -S_{m-1}+pS_{m-2}+S_{m-3}\\
					S_1 & S_2-2pS_1& \cdots & -S_{m-3}+pS_{m-4}+S_{m-5}& -S_{m-2}+pS_{m-3}+S_{m-4}\\
					0 & S_1& \cdots & -S_{m-4}+pS_{m-5}+S_{m-6}&-S_{m-3}+pS_{m-4}+S_{m-5}\\
					\vdots & \vdots & & \vdots & \vdots\\
					0&0 &\cdots& W_{p,n} & W_{p,n+1}-S_1\\
					0 &0 &  \cdots &W_{p,n-1}& W_{p,n}\\
			\end{vmatrix} \\
			&=\det(\Omega).\nonumber  
\end{align*}
  \end{minipage}
}

\medskip

	Let $a_{ij}$ denote the elements of the first $m-2$ column of $\Omega$  and let $b_{i,j}$   denote   the elements of the last two columns, i.e., $\Omega=\{a_{i,j},b_{i,m-1},b_{i,m}\}_{\underset{1\leq j \leq m-2}{1\leq i \leq m}}$.

\medskip

	The elements $a_{i,j}$ of   $\Omega$ satisfy $a_{i,j}=a_{i+1,j+1},\; (1\leq i\leq m-1,\;1\leq j\leq m-3),$ where
	\begin{displaymath}  
		\begin{cases}		  
			&a_{1,1}=S_2-2pS_1,\\  
			&a_{1,2}=S_3-2pS_2+(p^2-2)S_1,\\  
			&a_{1,3}=S_4-2pS_3+(p^2-2)S_2+2pS_1,\\  
			&a_{1,j}=S_{j+1}-2pS_{j}+(p^2-2)S_{j-1}+2pS_{j-2}+S_{j-3}, \quad 4\leq j \leq m-2;\\
			&a_{2,1}=S_1,\\
			&a_{i,1}=0, \quad 3\leq i \leq m.
		\end{cases}
	\end{displaymath}
	The elements $b_{i,j}$ of   $\Omega$ satisfy $b_{i,m-1}=b_{i+1,m},\; (1\leq i \leq m-3)$, where
	\begin{displaymath}
		\begin{cases}
			&b_{i,m-1}=-S_{m-i-1}+pS_{m-i-2}+S_{m-i-3},\quad 1\leq i \leq m-4;\\
			&b_{m-3,m-1}=-S_2+pS_1,\\
			&b_{m-2,m-1}=-S_1,\\
			&b_{m-1,m-1}=W_{p,n},\\
			&b_{m,m-1}=W_{p,n-1},	\\	
			&b_{1,m}=-S_{m-1}+pS_{m-2}+S_{m-3},\\
			&b_{m-1,m}=W_{p,n+1}-S_1,\\
			&b_{m,m}=W_{p,n}.
		\end{cases}
	\end{displaymath}	
	Now we use the following column transformations in the last determinant:
	\begin{align*}
		&b_{i,m-1}\longrightarrow b'_{i,m-1}:= b_{i,m-1}+\sum_{j=1}^{m-3}a_{i,j},\quad (1\leq i \leq m),\\
		&b_{i,m}\longrightarrow b'_{i,m}:= b_{i,m}+\sum_{j=1}^{m-2}a_{i,j},\quad (1\leq i \leq m),
	\end{align*}
	we obtain
	\begin{align}\label{Main9}
		\det\bigl( T_{m,n}^{(m-1)}\bigr) &=
		\begin{vmatrix}
				a_{1,1} & a_{1,2}&  \cdots& a_{1,m-2} &b'_{1,m-1}& b'_{1,m}\\
				a_{2,1} & a_{2,2}&\cdots & a_{2,m-2} &b'_{2,m-1} &b'_{2,m} \\
				\vdots & \vdots && \vdots & \vdots&\vdots\\
				0&0 &\cdots& a_{n-2,n-2}&b'_{m-2,m-1}&b'_{m-2,m}\\
				0&0 &\cdots& S_1 & W_{p,n}&W_{p,n+1}\\
				0 &0 &  \cdots &0& W_{p,n-1}&W_{p,n}
		\end{vmatrix},
	\end{align}
	where the last two columns are
	\begin{displaymath}
		\resizebox{1.01\textwidth}{!}{$
			\begin{pmatrix}
			(p^2-1)S_1+p^2\displaystyle\sum_{j=2}^{m-6}S_j	+(p^2-1)S_{m-5}+(p^2-2p)S_{m-4}	+(1-p)S_{m-3} &	(p^2-1)S_1+p^2\displaystyle\sum_{j=2}^{m-5}S_j+(p^2-1)S_{m-4}+(p^2-2p)S_{m-3}+(1-p)S_{m-2}	 \\
			p^2\displaystyle\sum_{j=1}^{m-7}S_j+(p^2-1)S_{m-6}+(p^2-2p)S_{m-5}+(1-p)S_{m-4}& p^2\displaystyle\sum_{j=1}^{m-6}S_j+(p^2-1)S_{m-5}+(p^2-2p)S_{m-4}+(1-p)S_{m-3}\\
			\hspace*{3cm}\vdots & \hspace*{3cm}\vdots \\
			p^2(S_1+S_2)+(p^2-1)S_{3}+(p^2-2p)S_{4}+(1-p)S_{5}& p^2(S_1+S_2+S_3)+(p^2-1)S_{4}+(p^2-2p)S_{5}+(1-p)S_{6}\\
			
			p^2S_1+(p^2-1)S_{2}+(p^2-2p)S_{3}+(1-p)S_{4}& p^2(S_1+S_2)+(p^2-1)S_{3}+(p^2-2p)S_{4}+(1-p)S_{5}\\
			
			(p^2-1)S_{1}+(p^2-2p)S_{2}+(1-p)S_{3}& p^2S_1+(p^2-1)S_{2}+(p^2-2p)S_{3}+(1-p)S_{4}\\
			
			(p^2-2p)S_{1}+(1-p)S_{2}& (p^2-1)S_{1}+(p^2-2p)S_{2}+(1-p)S_{3}\\
			(1-p)S_{1}& (p^2-2p)S_{1}+(1-p)S_{2}\\
			0&(1-p)S_1\\
			W_{p,n}&W_{p,n+1}\\
			W_{p,n-1}&W_{p,n}
			\end{pmatrix}$.}
	\end{displaymath}
	Furthermore, we perform the following column transformations on \eqref{Main9}:
	\begin{align*}
		&b'_{i,m-1}\longrightarrow b''_{i,m-1}:= b'_{i,m-1}+\sum_{j=1}^{m-4}(F_{p,m-j-2}-1)a_{i,j},\quad (1\leq i \leq m),\\
		&b'_{i,m}\longrightarrow b''_{i,m}:= b'_{i,m}+\sum_{j=1}^{m-3}(F_{p,m-j-1}-1)a_{i,j},
		\quad (1\leq i \leq m),
	\end{align*}
	where $F_{p,n}$ is the $n^{\rm th}$ $p$-Fibonacci number, we get
	\begin{align*}
		\det\bigl( T_{m,n}^{(m-1)}\bigr) &=
		 \begin{vmatrix}
				a_{1,1} & a_{1,2}&  \cdots& a_{1,m-2} &-S_1F_{p,m-2}& -S_1F_{p,m-1}\\
				a_{2,1} & a_{2,2}&\cdots & a_{2,m-2} &0 &0 \\
				%	0 &  a_{3,2}& \cdots &  a_{3,m-2}&0&0\\
				\vdots & \vdots && \vdots & \vdots&\vdots\\
				0 & 0&\cdots & a_{m-2,m-2} &0 &0 \\
				0&0 &\cdots& S_1 & W_{p,n}&W_{p,n+1}\\
				0 &0 &  \cdots &0& W_{p,n-1}&W_{p,n}\\
		\end{vmatrix}.
	\end{align*}
Next, we apply \eqref{cassini} with the following computational techniques on two last rows: 	
	{\small
		\begin{align*}
			\det\bigl( T_{m,n}^{(m-1)}\bigr)
			&= \frac{1}{W_{p,n-1}}
			\begin{vmatrix}
					%&\cdots & a_{1,m-2}&-S_1F_{p,m-2}& -S_1F_{p,m-1}\\
					& & \vdots &\vdots&\vdots\\
					&\cdots& S_1 & W_{p,n}&W_{p,n+1}\\
					& \cdots &0& W^2_{p,n-1}&W_{p,n}W_{p,n-1}\\
			\end{vmatrix}
			\\
			&= \frac{1}{W_{p,n-1}}
		\begin{vmatrix}
					%&\cdots & a_{1,m-2}&-S_1F_{p,m-2}& -S_1F_{p,m-1}\\
					& & \vdots &\vdots&\vdots\\
					&\cdots& S_1 & W_{p,n}&W_{p,n+1}\\
					& \cdots &-S_1W_{p,n-2}& (-1)^{n-1}\Delta&p(-1)^{n-1}\Delta\\
			\end{vmatrix} 
			\\
			&=\frac{(-1)^{n-1}}{\Delta W_{p,n-1}}
			\begin{vmatrix}
					%\cdots & a_{1,m-2}&-S_1F_{p,m-2}& -S_1F_{p,m-1}\\
					& \vdots &\vdots&\vdots\\
					\cdots & (-1)^{n-1}\Delta S_1 & (-1)^{n-1}\Delta W_{p,n}&(-1)^{n-1}\Delta W_{p,n+1}\\
					\cdots &-S_1W_{p,n-2}& (-1)^{n-1}\Delta&p(-1)^{n-1}\Delta\\
			\end{vmatrix}
			\\
			&=\frac{(-1)^{n-1}}{\Delta}
		\begin{vmatrix}
					a_{1,1} & \cdots& a_{1,m-2} &-S_1F_{p,m-2}& -S_1F_{p,m-1}\\
					a_{2,1} &\cdots & a_{2,m-2} &0 &0 \\
					\vdots && \vdots & \vdots&\vdots\\
					0&\cdots& S_1W_{p,n-1} &0&(-1)^{n-1}\Delta\\
					0 & \cdots &-S_1W_{p,n-2}& (-1)^{n-1}\Delta&(-1)^{n-1}p\Delta\\
			\end{vmatrix}.
		\end{align*}
	}
	Thus, we have
		\begin{align*}
			\det\bigl( T_{m,n}^{(m-1)}\bigr)
			&=\frac{(-1)^{n}}{\Delta}
			\begin{vmatrix}
					a_{1,1} & \cdots& a_{1,m-2} &-S_1F_{p,m-3}& -S_1F_{p,m-2}\\
					a_{2,1} &\cdots & a_{2,m-2} &0 &0 \\
					\vdots && \vdots & \vdots&\vdots\\
					0&\cdots& S_1W_{p,n-1}&(-1)^{n-1}\Delta&0\\
					0 & \cdots &-S_1W_{p,n-2}& 0&(-1)^{n-1}\Delta\\
			\end{vmatrix}\\
			&=\frac{(-1)^{n}}{\Delta}
			\begin{vmatrix}
					a_{1,1} & \cdots& A &-S_1F_{p,m-3}& -S_1F_{p,m-2}\\
					a_{2,1} &\cdots & a_{2,m-2} &0 &0 \\
					\vdots && \vdots & \vdots&\vdots\\
					0&\cdots&0&(-1)^{n-1}\Delta&0\\
					0 & \cdots &0& 0&(-1)^{n-1}\Delta\\
			\end{vmatrix} \\
			&=(-1)^{n}\Delta
			\begin{vmatrix}
					a_{1,1} &a_{1,2} & \cdots& a_{1,m-3}&A \\
					S_1 &a_{2,2} &\cdots & a_{2,m-3}&a_{2,m-2} \\
					0 & S_1 & \cdots &  a_{3,m-3}&a_{3,m-2}\\
					\vdots&\vdots &  &\vdots\\
					0&0&\cdots& S_1&a_{m-2,m-2}\\
			\end{vmatrix},
	\end{align*}
	where
	\begin{align}
		A&=a_{1,m-2}+\frac{(-1)^nS_1^2}{\Delta}(F_{p,m-2}W_{p,n-2}-F_{p,m-3}W_{p,n-1})\nonumber\\
		&=a_{1,m-2}+\frac{(-1)^nS_1^2}{\Delta}(-1)^{m-1}W_{p,n-m+1}\quad\text{(using \eqref{EQQ})}.\label{EQA}
	\end{align}
	The last determinant can be rewritten as follows:
	\begin{align}
		\det\bigl( T_{m,n}^{(m-1)}\bigr)&=(-1)^{n}\Delta\left((-1)^{m-1}S_{1}^{m-3}A+\sum_{j=1}^{m-3}(-1)^{i+1}a_{1,j}M_{1,j}\right),
	\end{align}
	where $M_{1,j}$ denotes the determinant obtained from $\det\bigl(T_{m,n}^{(m-1)}\bigr)$ by omitting the first
	row and $j^{\rm th}$ column $(1\leq j \leq m-3)$.
	Using Remark \ref{rem1}, we get
	$a_{i,j}=p^{n}f_{i,j}(n)$ where $f_{i,j}(n)$ is a polynomial in $n$. This implies that $M_{1,j}=p^{n(m-3)}f_j(n)$ where $f_{j}(n)$ is a polynomial on $n$ $(1\leq j\leq m-3)$. Therefore, we have
	\begin{align*}	
		\det\bigl(T_{m,n}^{(m-1)}\bigr)&=(-1)^{n}\Delta((-1)^{m-1}S_{1}^{m-3}A+p^{n(m-2)}f(n))\quad (S_1:=bp^{n})\\
		&=p^{n(m-2)}b^{m-1}(p^{n}W_{p,n-m+1}+g(n)) \quad \text{(using \eqref{EQA})},
	\end{align*}
	where $f(n)$ and $g(n)$ are polynomials in $n$.	The existence of a positive integer $n_m$ such that $W_{p,n-m+1}>P(n)$ for all $n > n_m$, where $P(n)$ is a polynomial of any degree, completes the proof.
\end{proof}
\begin{corollary}\label{Theorem1}
	Let $r\geq 1$  be an integer. For every integer $m$ such that $1\leq m\leq r+1$, there exists a positive integer $n_m$ such that  $\det\bigl( T_{m,n}^{(r)}\bigr) >0$ for all $n\geq n_m$.
\end{corollary}
\begin{proof}
We use induction on $r$. The case $r=m-1$ is given in Theorem \ref{lemma}. Assume that the claim is true for all $m-1\leq t \leq r-1$; we prove that 
	\begin{align*}
		\det\bigl( T_{m,n}^{(r)}\bigr)=
		\begin{vmatrix}
				W_{p,n}^{(r)} & W_{p,n-1}^{(r)} & \cdots & W_{p,n-m+1}^{(r)}\\
				W_{p,n+1}^{(r)} & W_{p,n}^{(r)} & \cdots & W_{p,n-m+2}^{(r)}\\
				\vdots & \vdots & \ddots & \vdots\\
				W_{p,n+m-1}^{(r)} & W_{p,n+m-2}^{(r)} & \cdots & W_{p,n}^{(r)}\\
		\end{vmatrix} >0.
	\end{align*}
	By performing row transformations using relation \eqref{Id2}, we obtain
	\begin{align*}
		\det\bigl( T_{m,n}^{(r)}\bigr)  &=
	\begin{vmatrix}
				W_{p,n}^{(r)} & W_{p,n-1}^{(r)} & \cdots & W_{p,n-m+1}^{(r)}\\
				W_{p,n+1}^{(r-1)} & W_{p,n}^{(r-1)} & \cdots & W_{p,n-m+2}^{(r-1)}\\
				\vdots & \vdots & \ddots & \vdots\\
				W_{p,n+m-1}^{(r-1)} & W_{p,n+m-2}^{(r-1)} & \cdots & W_{p,n}^{(r-1)}\\
		\end{vmatrix}  \\
		&=\sum_{j=1}^{m}(-1)^{j+1}\frac{W^{(r)}_{p,n-j+1}}{W^{(r-1)}_{p,n-j+1}}W^{(r-1)}_{p,n-j+1}M_{1,j},
	\end{align*}
	where $M_{1,j}$ denotes the determinant obtained from $\det\bigl(T_{m,n}^{(r)}\bigr)$ by omitting the first
	row and $j^{\rm th}$ column $(1\leq j \leq m)$.
	We define a function $f:\mathbb{R}^{n}\longrightarrow \mathbb{R}$ by
	\begin{equation*}
		f(x_1,x_2,\dots,x_n)=\sum_{j=1}^{m}(-1)^{j+1}x_jW^{(r-1)}_{p,n-j+1}M_{1,j}.
	\end{equation*}
	Let $\phi:=1+p\alpha$, where $\alpha=\frac{p+\sqrt{p^2+4}}{2}$. We have $f(\phi,\phi,\dots,\phi)=\phi\sum_{j=1}^{m}(-1)^{j+1}W^{(r-1)}_{p,n-j+1}M_{1,j}=\phi\det\bigl( T_{m,n}^{(r-1)}\bigr) >0$.
	Since  $f$ is continuous, there is a neighborhood  
	$$I=(\phi-\epsilon_1,\phi+\epsilon_1)\times(\phi-\epsilon_2,\phi+\epsilon_2) \times\cdots \times  (\phi-\epsilon_n,\phi+\epsilon_n)$$
	of the point
	$(\phi,\phi,\dots,\phi)$ such that $f$ is positive on $I$. 
	Using \eqref{Id3}, by dividing both sides  by  $W_{n}^{(r-1)}$, and
	passing to limit $n\longrightarrow \infty $, one readily obtains
	\begin{align*}
		\lim_{n\to \infty}\frac{W_{p,n}^{(r)}}{W_{p,n}^{(r-1)}}=	\lim_{n\to \infty}\frac{W_{p,n+2r}}{W_{p,n+2r-2}}=\phi.
	\end{align*}
	This implies that, for sufficiently large  $n$, the coefficient $x_j=\frac{W^{(r)}_{p,n-j+1}}{W^{(r-1)}_{p,n-j+1}}$ falls into $(\phi-\epsilon_j,\phi+\epsilon_j)$ for all $1\leq j\leq n$. Therefore, there exists a positive integer $n_m$ such that 
	\begin{align*}
		\det\bigl(T_{m,n}^{(r)}\bigr)=	f\left(\frac{W^{(r)}_{p,n}}{W^{(r-1)}_{p,n}},\frac{W^{(r)}_{p,n-1}}{W^{(r-1)}_{p,n-1}},\dots,\frac{W^{(r)}_{p,n-m+1}}{W^{(r-1)}_{p,n-m+1}}\right) >0
	\end{align*}
for all $n\geq n_m$.
	The proof is thus completed.
\end{proof}

If we take $m=2$ in Corollary \ref{Theorem1}, we obtain
\begin{corollary}\label{Tota_positi_W}
	Let $r\geq 1$ be an integer.
	The  sequence $(W_{p,n}^{(r)})_{n}$ is log-concave for large enough $n$.
\end{corollary}

\begin{corollary}
	Let $r\geq 1$ be an integer.
	The  sequences $(F_{p,n}^{(r)})_{n}$  and  $(L_{p,n}^{(r)})_{n}$ are log-concave for large enough $n$.
\end{corollary}

For $p=1$, we obtain the log-concavity of the sequences $(F_n^{(r)})_n$ and   $(L_n^{(r)})_n$.
This recovers the main result established by Zheng et al.\ \cite{Log-ConcavityoftheHyperfibonacci}.

\section{Total positivity of Toeplitz matrices}\label{Sec4}
Let $T_{r+2,n}^{(r)}$ denote the Toeplitz matrix of order $r+2$ consisting of generalized hyper-Fibonacci numbers of the $r^{\rm th}$ generation; that is
\begin{align*}
	T_{r+2,n}^{(r)} =
\begin{pmatrix}
			W_{p,n}^{(r)} & W_{p,n-1}^{(r)} & \cdots & W_{p,n-r-1}^{(r)}\\
			W_{p,n+1}^{(r)} & W_{p,n}^{(r)} & \cdots & W_{p,n-r}^{(r)}\\
			\vdots & \vdots & \ddots & \vdots\\
			W_{p,n+r+1}^{(r)} & W_{p,n+r}^{(r)} & \cdots & W_{p,n}^{(r)}\\
	\end{pmatrix}.
\end{align*}
Let $\Delta:=a^2+pab-b^2$. Then we have the following main theorem.
\begin{theorem}\label{Theo total positivity}
	Let $r\geq 1$ be an integer. 
	\begin{enumerate}
		\item 	If $\Delta>0$, then there exists a positive integer $n_r$ such that the matrix $T_{r+2,2n}^{(r)}$ is totally positive for every $n\geq n_r$.
		\item If $\Delta<0$, then there exists a positive integer $n'_r$ such that the matrix $T_{r+2,2n+1}^{(r)}$ is totally positive for every $n\geq n'_r
		$.
	\end{enumerate}
\end{theorem}
\begin{proof}
	The $2r+1$ initial minors of order two of $T_{r+2,n}^{(r)}$ have the form $T_{2,n_2}^{(r)}$, where $n_2\geq  n-r$. 	Using Corollary \ref{Theorem1}, there exists a positive integer $q_2$ such that these initial minors 
	are positive for $n_2\geq q_2$. Similarly, according to Corollary \ref{Theorem1}, there exist positive integers $q_3,q_4,\dots,q_{r+1}$ such that the initial minors of order $3,4,\dots,r+1$, which are of the form $T_{3,n_3}^{(r)}$,$T_{4,n_4}^{(r)}$,\dots, $T_{r+1,n_{r+1}}^{(r)}$, 
	are positive for $n_3\geq q_3$, $n_4\geq q_4, \ldots, n_{r+1}\geq q_{r+1}$. It remains to discuss the positivity of $\det\bigl(T_{r+2,n}^{(r)}\bigr) $.
	By reversing the order of columns of $T_{r+2,n}^{(r)}$, we obtain 
	\begin{align*}
		\det\bigl( T_{r+2,n}^{(r)}\bigr)&=(-1)^{\lfloor \frac{r+2}{2}\rfloor}
	\begin{vmatrix}
				W_{p,n-r-1}^{(r)}	&W_{p,n-r}^{(r)}&\cdots &W_{p,n}^{(r)}\\
				W_{p,n-r}^{(r)}	&W_{p,n-r+1}^{(r)}&\cdots&W_{p,n+1}^{(r)} \\
				\vdots & \vdots & \ddots & \vdots\\
				W_{p,n}^{(r)}&W_{p,n+1}^{(r)}& \cdots& W_{p,n+r+1}^{(r)}
		\end{vmatrix}  \\
		&=(-1)^{\lfloor \frac{r+2}{2}\rfloor}\det(H_{r,n-r-1})\quad (\text{using Proposition \ref{propo2}})\\
		&=(-1)^{n}p^{(n-1)r}b^{r}\Delta.
	\end{align*}
\end{proof}
The following corollary follows directly from Theorem \ref{Theo total positivity}.
\begin{corollary}\label{Cor_total positivity}Let $r\geq 1$ be a positive integer. There exists a positive integer $n_r$ such that the matrix $T_{r+2,n}^{(r)}$ 	is $\text{TP}_{r+1}$ for all $n\geq n_r$.
\end{corollary}

Furthermore, the following corollaries follow directly from Theorem \ref{Theo total positivity} and Corollary \ref{Cor_total positivity}.

\begin{corollary}
	Let $r\geq 1$ be a positive integer. There exist positive integers $n_r$ and $n'_r$ such that
	\begin{enumerate}
		\item The matrix
		\begin{align*}
			\mathcal{F}_{r+2,2n+1} =
		\begin{pmatrix}
					F_{p,2n+1}^{(r)} & F_{p,2n}^{(r)} & \cdots & F_{p,2n-r}^{(r)}\\
					F_{p,2n+2}^{(r)} & F_{p,2n+1}^{(r)} & \cdots & F_{p,2n-r+1}^{(r)}\\
					\vdots & \vdots & \ddots & \vdots\\
					F_{p,2n+r+2}^{(r)} & F_{p,2n+r+1}^{(r)} & \cdots & F_{p,2n+1}^{(r)}\\
			\end{pmatrix}	
		\end{align*}
		is totally positive for every $n\geq n_r
		$.
		\item The matrix
		\begin{align*}
			\mathcal{F}_{r+2,n} =
		\begin{pmatrix}
					F_{p,n}^{(r)} & F_{p,n-1}^{(r)} & \cdots & F_{p,n-r-1}^{(r)}\\
					F_{p,n+1}^{(r)} & F_{p,n}^{(r)} & \cdots & F_{p,n-r}^{(r)}\\
					\vdots & \vdots & \ddots & \vdots\\
					F_{p,n+r+1}^{(r)} & F_{p,n+r}^{(r)} & \cdots & F_{p,n}^{(r)}\\
			\end{pmatrix} 	
		\end{align*}
		
		is $\text{TP}_{r+1}$ for every $n\geq n'_r
		$.
	\end{enumerate}
\end{corollary}
For $p=1$, we obtain the main results given by Do\v{s}li\'c et al.\ \cite{Martinjak}.

\begin{corollary}
	Let $r\geq 1$ be an integer. There exist positive integers $n_r$ and $n'_r$ such that:
	\begin{enumerate}
		\item The matrix
		\begin{align*}
			\mathcal{L}_{r+2,2n} =
			\begin{pmatrix}
					L_{p,2n}^{(r)} & L_{p,2n-1}^{(r)} & \cdots & L_{p,2n-r-1}^{(r)}\\
					L_{p,2n+1}^{(r)} & L_{p,2n}^{(r)} & \cdots & L_{p,2n-r}^{(r)}\\
					\vdots & \vdots & \ddots & \vdots\\
					L_{p,2n+r+1}^{(r)} & L_{p,2n+r}^{(r)} & \cdots & L_{p,2n}^{(r)}\\
			\end{pmatrix} 
		\end{align*}
		is totally positive for every $n\geq n_r
		$.
		\item The matrix
		\begin{align*}
			\mathcal{L}_{r+2,n} =
			\begin{pmatrix}
					L_{p,n}^{(r)} & L_{p,n-1}^{(r)} & \cdots & L_{p,n-r-1}^{(r)}\\
					L_{p,n+1}^{(r)} & L_{p,n}^{(r)} & \cdots & L_{p,n-r}^{(r)}\\
					\vdots & \vdots & \ddots & \vdots\\
					L_{p,n+r+1}^{(r)} & L_{p,n+r}^{(r)} & \cdots & L_{p,n}^{(r)}\\
			\end{pmatrix} 
		\end{align*}
	\end{enumerate}
	is $\text{TP}_{r+1}$ for every $n\geq n'_r$.
\end{corollary}
For $p=1$, we obtain the main results given by Rezig and Ahmia \cite{AhmiaRezig}.

\section{Final remarks}
This work relies on specific choices of coefficients, initial conditions, and parity assumptions. A complete characterization of total positivity without these constraints remains open and will likely require new techniques.

\section{Acknowledgments}
We thank the reviewer and the editor for their valuable remarks and suggestions, which have improved the quality of the paper.
	
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	\noindent 2020 {\it Mathematics Subject Classification}:
	Primary 15B36; Secondary 11B39, 15A45.
	
	\noindent \emph{Keywords: }hyper-{F}ibonacci number, hyper-{L}ucas number, {T}oeplitz matrix, total positivity, log-concavity.

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\noindent (Concerned with sequences
\seqnum{A000032},
\seqnum{A000045},
\seqnum{A000129}, and
\seqnum{A001333}.)


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\vspace*{+.1in}
\noindent
Received  May 12 2025;
revised versions received   May 14 2025; February 26 2026.
Published in {\it Journal of Integer Sequences}, April 22 2026.
	
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Return to \href{https://cs.uwaterloo.ca/journals/JIS/}{Journal of Integer Sequences home page}.
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