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\begin{center}
\vskip 0.5cm{\large\bf 
Permanental Representations of Negatively Subscripted \\
\vskip .05in
Generalized Order-$k$ Fibonacci Numbers
}
\vskip 0.5cm
Hac\`{e}ne Belbachir \\
USTHB \\ Faculty of Mathematics \\
RECITS Laboratory\\
P. O. Box 32\\ El Alia, 16111, Algiers\\
Algeria\\
\href{mailto:hbelbachir@usthb.dz}{\tt hbelbachir@usthb.dz}\\
\ \\
Ihab-Eddine Djellas \\
USTHB \\ Faculty of Mathematics \\
RECITS Laboratory\\
P. O. Box 32 \\
El Alia, 16111, Algiers\\
Algeria\\
Centre de Recherche sur l'Information Scientifique et Technique--CERIST\\
Algiers, Algeria \\
\href{mailto:idjellas@usthb.dz}{\tt idjellas@usthb.dz}\\
\ \\
Fariza Krim\\ 
Oran 1 University \\ Faculty of Sciences\\ GEAN Laboratory\\
P. O. Box 1524 \\
El Menaouer, 31000, Oran \\
Algeria
USTHB \\ Faculty of Mathematics \\ RECITS Laboratory\\
P. O. Box 32 \\
El Alia, 16111, Algiers \\
Algeria\\
\href{mailto:fkrim@usthb.dz}{\tt fkrim@usthb.dz}
\end{center}

\begin{abstract}
We introduce an extension to the negative indices of a family of
generalized Fibonacci sequences of order-$k$, and for which we establish
recurrence relations. We also give permanental representations of
negatively subscripted generalized Fibonacci and Lucas sequences via
Hessenberg matrices.
\end{abstract}


\section{Introduction}

A family of $k$ sequences of the generalized order-$k$ Fibonacci numbers was defined and studied by Er \cite{er}, for $n>0$ and $1 \leq i \leq k$, as follows:
\begin{equation}
g^{(i)}_{n}= c_{1}g^{(i)}_{n-1}+c_{2}g^{(i)}_{n-2}+\cdots+c_{k}g^{(i)}_{n-k}, \label{k_fibo}
\end{equation}
where $c_{1},c_{2},\dots,c_{k}$ are real constant coefficients, with $c_{k}\neq 0$, and for $1-k \leq n \leq 0$,
\begin{displaymath}
g^{(i)}_{n} = \begin{cases}
        1, & \text{for $i=1-n$;}\\
        0, & \text{otherwise.}
\end{cases}
\end{displaymath}

Several sequences were derived from \eqref{k_fibo}. Lee and Lee \cite{ll} studied the sequence obtained from \eqref{k_fibo} by setting $(c_{1},c_{2},\dots,c_{k})=(1,1,\dots,1)$ and $i=1$, which is called in the present paper order-$k$ Fibonacci sequence $(\widetilde{g}^{(k)}_{n})_{n}$. Also, Lee \cite{G.Lee} studied order-$k$ Lucas sequence denoted $(l_{n}^{(k)})_{n}$, for $k\geq 2$. The sequence $(l_{n}^{(k)})_{n}$ is obtained from \eqref{k_fibo} by setting $(c_{1},c_{2},\dots,c_{k})=(1,1,\dots,1)$ with initial conditions $l_{0}^{(k)}=2$, $l_{j}^{(k)}=2^{j-1}$, for $j=1,\ldots ,k-1$, and $l_{k}^{(k)}=2^{k-1}+1$.

In the present paper, we consider the family of shifted generalized order-$k$ Fibonacci sequences defined as \eqref{k_fibo}, but with initial conditions, for $0 \leq n \leq k-1$,
\begin{displaymath}
g^{(i)}_{n} = \begin{cases}
        1, & \text{for $i = k - n$;}\\
        0, & \text{otherwise.}
\end{cases}
\end{displaymath}

The \textit{permanent} of a square matrix 
$A=(a_{ij})_{n\times n}$ is defined as follows:
\begin{equation*}
\per A=\sum_{\sigma \in \mathfrak{S}_{n}}\prod_{i=1}^{n}a_{i\sigma(i)},
\end{equation*}
where the summation runs over all permutations $\sigma$ of the symmetric group $\mathfrak{S}_{n}$ of order $n$. 

Next, we present the definition of matrix contraction; see for instance \cite{R.A.Brualdi}. Let $A=(a_{ij})$ be an $m\times n$ real matrix with row vectors $\alpha _{1},\alpha _{2},\ldots ,\alpha _{m}$. We say that $A$ is \textit{contractible} on column (resp., row) $k$ if column (resp., row) $k$ contains exactly two nonzero entries. If $A$ is contractible on column $k$ with $a_{ik}\neq 0\neq a_{jk}$ and $i\neq j$, then the $(m-1)\times (n-1)$ matrix $A_{ij}^{(k)}$, obtained from $A$ by replacing row $i$ by $a_{jk}\alpha
_{i}+a_{ik}\alpha _{j}$ and deleting row $j$ and column $k$, is called the
contraction of $A$ on column $k$ relative to rows $i$ and $j$.

Brualdi and Gibson \cite{R.A.Brualdi} gave the following lemma.
\begin{lemma}
\label{Brualdi}Let $A$ be a nonnegative integral matrix of order $n>1$, and
let $B$ be a contraction of $A$. Then
\begin{equation*}
\per A= \per B.
\end{equation*}
\end{lemma}

Permanental representations of recurrent sequences have been investigated by several authors. Yilmaz and Bozkurt \cite{yilmaz_2} gave permanental representations of sums of Fibonacci and Lucas numbers via Hessenberg matrices. Lee et al.~\cite{lls} obtained $g_{n+k-1}^{(k)}$ as the permanent of an $n\times n$ upper triangular $(0,1)$-matrix; that is a matrix whose all entries are either $0$ or $1$. Lee \cite{G.Lee} constructed a $(0,1)$-matrix of order $n$ and established that the permanent of that matrix is the $(n-1)^{\text{th}}$ term $l_{n-1}^{(k)}$ of the order-$k$ Lucas sequence.

The aim of this work is to propose an extension to negative indices of the family of generalized order-$k$ Fibonacci sequences $(g^{(i)}_{n})_{n}$, $1 \leq i \leq k$. For instance, extensions to negative indices of order-$k$ Fibonacci, order-$k$ Lucas, order-$k$ Pell, and order-$k$ Jacobsthal are also given. As application, we provide some permanental representations. 

The present paper is organized as follows: In Section \ref{extension to negative}, we give an extension of order-$k$ Fibonacci, order-$k$ Lucas, and the family of generalized order-$k$ Fibonacci sequences $(g^{(i)}_{n})_{n}$, $1 \leq i \leq k$, to negatively subscripted indices. We also provide recurrence formulas for the $n^{\text{th}}$ term of these negatively subscripted sequences. In Section \ref{representations}, we present negatively subscripted generalized order-$k$ Fibonacci numbers as the permanent of special Hessenberg matrix. In Section \ref{more representations}, we give more permanental representations of negatively subscripted order-$k$ Fibonacci and order-$k$ Lucas sequences. We conclude by Section \ref{proofs}, where we provide proofs of the results presented in Sections \ref{representations} and \ref{more representations}.


\section{Extension to negative indices}\label{extension to negative}

Extension to negative indices of classical Fibonacci sequence $(F_{n})_{n}$ with initial conditions $F_{0}=0$, $F_{1}=1$ is given for $n\geq 1$, by
\begin{equation}
F_{-n}=F_{-n+2}-F_{-n+1}=(-1)^{n+1}F_{n}.
\end{equation}
The extension to negative indices of classical Lucas sequence $(L_{n})_{n}$ with initial conditions $L_{0}=2$, $L_{1}=1$ is given for $n\geq 1$, by
\begin{equation}
L_{-n}=L_{-n+2}-L_{-n+1}=(-1)^{n}L_{n}.
\end{equation}

Firstly, we propose an extension of order-$k$ Fibonacci sequence $(\widetilde{g}_{n}^{(k)})_{n}$ to negatively subscripted indices, for $n\geq 1$, as follows: 
\begin{equation}
\widetilde{g}_{-n}^{(k)}=-\widetilde{g}_{-n+1}^{(k)}-\widetilde{g}_{-n+2}^{(k)}-\cdots -\widetilde{g}_{-n+k-1}^{(k)}+\widetilde{g}_{-n+k}^{(k)}.\label{negafib}
\end{equation}

For example, for $k=2$, $(\widetilde{g}_{n}^{(2)})_{n\in \mathbb{Z}}$ is the classical Fibonacci sequence.

For $k=3$, $(\widetilde{g}_{n}^{(3)})_{n\in \mathbb{Z}}$ is the Tribonacci sequence
$$\ldots ,-8,4,1,-3,2,0,-1,1,0,0,1,1,2,4,7,13,\ldots $$

For $k=4$, $(\widetilde{g}_{n}^{(4)})_{n\in \mathbb{Z}}$ is the Quadrabonacci sequence
$$\ldots,0,1,-3,2,0,0,-1,1,0,0,0,1,1,2,4,8,15,29,\ldots $$

An extension of order-$k$ Lucas sequence $(l_{n}^{(k)})_{n}$ to negatively subscripted indices, for $n\geq 1$, is given by
\begin{equation}
l_{-n}^{(k) }=-l_{-n+1}^{(k)}-l_{-n+2}^{(k)}-\cdots -l_{-n+k-1}^{(k)}+l_{-n+k}^{(k)}. \label{negalucas}
\end{equation}

For example, for $k=2$, $(l_{n}^{(2)})_{n\in \mathbb{Z}}$ is the Lucas sequence.

For $k=3$, $(l_{n}^{(3)})_{n\in \mathbb{Z}}$ is the Tribonacci-Lucas sequence
$$\ldots, 6,-11,6,1,-4,3,0,-1,2,1,2,5,8,15,28,51,\ldots$$
 
For $k=4$, $(l_{n}^{(4)})_{n\in \mathbb{Z}}$ is the Quadrabonacci-Lucas sequence
$$\ldots ,-11,6,0,1,-4,3,0,0,-1,2,1,2,4,9,16,31,60,\ldots$$

Secondly, we establish the extension of the family of generalized order-$k$ Fibonacci sequences, $(g^{(i)}_{n})_{n}$, $1 \leq i \leq k$, to negatively subscripted indices, in the following way:
\begin{equation}
g^{(i)}_{-n}= -\frac{c_{k-1}}{c_{k}}g^{(i)}_{-n+1}-\frac{c_{k-2}}{c_{k}}g^{(i)}_{-n+2}-\cdots-\frac{c_{1}}{c_{k}}g^{(i)}_{-n+k-1}+\frac{1}{c_{k}}g^{(i)}_{-n+k}. \label{neg_k_fibo}
\end{equation}
where, $c_{j}\in \mathbb{R}$ for $1\leq j\leq k$ and $c_{k} \neq 0$. And $g^{(i)}_{-n}$ is said to be the $n^{\text{th}}$ negatively subscripted generalized order-$k$ Fibonacci number of the sequence number $i$ of the family.

For instance, we give the extension to negative indices of classical order-$k$ sequences. We start by a family of order-$k$ Fibonacci sequences defined for $1\leq i \leq k$ and $n \geq k$ as follows:
\begin{equation*}
f^{(i)}_{n}= f^{(i)}_{n-1}+f^{(i)}_{n-2}+\cdots+f^{(i)}_{n-k},
\end{equation*}
with initial conditions; for $0 \leq n \leq k-1$,
\begin{displaymath}
f^{(i)}_{n} = \begin{cases}
        1, & \text{for $i = k - n$;}\\
        0, & \text{otherwise.}
\end{cases}
\end{displaymath}

For $n \geq 1$, we give the extension of the family of order-$k$ Fibonacci sequences to negative indices as follows:
\begin{equation} 
f^{(i)}_{-n}= -f^{(i)}_{-n+1}-f^{(i)}_{-n+2}-\cdots -f^{(i)}_{-n+k-1}+f^{(i)}_{-n+k}. \label{neg_k_fiboo}
\end{equation}
The following table gives the first terms of negatively subscripted order-$k$ Fibonacci sequences for $n\geq 1$,

\begin{center}
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{|l|c|c|c|}
\hline
       & $k=2$ & $k=3$ & $k=4$  \\\hline
$i=1$  & $1, -1, 2, -3, 5, -8, 13, -21, 34, -55 \ldots$ & $1, -1, 0, 2, -3, 1, 4, -8, 5, 7 \ldots$ & $1, -1, 0, 0, 2, -3, 1, 0, 4, -8 \ldots$\\ \hline
$i=2$ & $-1, 2, -3, 5, -8, 13, -21, 34, -55, 89 \ldots$ & $-1, 2, -1, -2, 5, -4, -3, 12, -13, -2 \ldots$ & $-1, 2, -1, 0, -2, 5, -4, 1, -4, 12 \ldots$\\ \hline
$i=3$ & & $-1, 0, 2, -3, 1, 4, -8, 5, 7, -20 \ldots $ &  $-1, 0, 2, -1, -2, 1, 4, -4, -3, 4 \ldots$\\ \hline
$i=4$ & & & $-1, 0, 0, 2, -3, 1, 0, 4, -8, 5 \ldots$\\ \hline
\end{tabular}
\end{adjustbox}
\captionof{table}{First terms of negatively subscripted order-$k$ Fibonacci sequences}
\end{center} 

\begin{remark}
For $k=2$, $(f^{(1)}_{-n})_{n}$ is the extension to negative indices of classical Fibonacci numbers.
\end{remark} 

Kili\c{c} and Ta\c{s}ci \cite{kilic_2} studied a family of $k$ sequences of order-$k$ Pell numbers defined for $n > 0$ and $1 \leq i \leq k$, as follows:
\begin{equation}
P^{(i)}_{n}= 2P^{(i)}_{n-1}+P^{(i)}_{n-2}+\cdots +P^{(i)}_{n-k}. \label{k_pell}
\end{equation}
with initial conditions; for $1-k \leq n \leq 0$,
\begin{displaymath}
P^{(i)}_{n} = \begin{cases}
        1, & \text{for $n = 1 - i$;}\\
        0, & \text{otherwise.}
\end{cases}
\end{displaymath}
Next, we consider $k$ sequences of shifted order-$k$ Pell numbers \eqref{k_pell} for $n \geq k$ and $1 \leq i \leq k$, with initial conditions, for $0 \leq n \leq k-1$,
\begin{displaymath}
P^{(i)}_{n} = \begin{cases}
        1, & \text{for $i = k - n$;}\\
        0, & \text{otherwise.}
\end{cases}
\end{displaymath}

Then for $n \geq 1$, we give extension of these sequences \eqref{k_pell} to negative indices as follows:
\begin{equation} 
P^{(i)}_{-n}= -P^{(i)}_{-n+1}-\cdots -P^{(i)}_{-n+k-2}-2P^{(i)}_{-n+k-1}+P^{(i)}_{-n+k}. \label{neg_k_pell}
\end{equation}
The next table gives the first terms of negatively subscripted order-$k$ Pell sequences for $n\geq 1$,
\begin{center}
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{|l|c|c|c|}
\hline
       & $k=2$ & $k=3$ & $k=4$  \\\hline
$i=1$  & $1, -2, 5, -12, 29, -70, 169, -408 \ldots$ & $1, -1, -1, 4, -3, -6, 16, -7, -31, 61 \ldots$ & $1, -1, 0, -1, 4, -4, 2, -7, 17, -18 \ldots$\\ \hline
$i=2$ & $-2, 5, -12, 29, -70, 169, -408, 985 \ldots$ & $-2, 3, 1, -9, 10, 9, -38, 30, 55, -153 \ldots$ & $-2, 3, -1, 2, -9, 12, -8, 16, -41, 53 \ldots$\\ \hline
$i=3$ & & $-2, 3, -3, 1, 4, -12, 21, -26, 19, 9 \ldots $ &  $-1, -1, 3, 0, -2, -5, 10, -1, -1, -23 \ldots$\\ \hline
$i=4$ & & & $-1, 0, -1, 4, -4, 2, -7, 17, -18, 17 \ldots$\\ \hline
\end{tabular}
\label{pell-table}
\end{adjustbox}
\captionof{table}{First terms of negatively subscripted order-$k$ Pell sequences}
\end{center}

\begin{remark}
Some of the sequences in Table \ref{pell-table} are known in OEIS \cite{sloane}, as for example, \seqnum{A215936}, \seqnum{A276229}, and \seqnum{A078021}.
\end{remark}

In 2009, Yilmaz and Bozkurt \cite{yilmaz} studied a family of $k$ sequences of order-$k$ Jacobsthal numbers defined for $n > 0$ and $1 \leq i \leq k$, as follows:
\begin{equation}
J^{(i)}_{n}= J^{(i)}_{n-1}+2J^{(i)}_{n-2}+\cdots +J^{(i)}_{n-k}, \label{k_jacob}
\end{equation}
with initial conditions for $1-k \leq n \leq 0$,
\begin{displaymath}
J^{(i)}_{n} = \begin{cases}
        1, & \text{for $n = 1 - i$;}\\
        0, & \text{otherwise.}
\end{cases}
\end{displaymath}
As before, we consider $k$ sequences of shifted order-$k$ Jacobsthal numbers \eqref{k_jacob} for $n \geq k$ and $1 \leq i \leq k$, with initial conditions for $0 \leq n \leq k-1$,
\begin{displaymath}
J^{(i)}_{n} = \begin{cases}
        1, & \text{for $i = k - n$;}\\
        0, & \text{otherwise.}
\end{cases}
\end{displaymath}

Then, for $n \geq 1$, we give the extension of these sequences \eqref{k_jacob} to negative indices as follows:
\begin{align}
J^{(i)}_{-n}& = -\frac{1}{2}J^{(i)}_{-n+1}+\frac{1}{2}J^{(i)}_{-n+2}, \text{   for } k=2 \text{,  } 1\leq i \leq 2,\\
J^{(i)}_{-n}& = -J^{(i)}_{-n+1}-\cdots -J^{(i)}_{-n+k-3}-2J^{(i)}_{-n+k-2}-J^{(i)}_{-n+k-1}+J^{(i)}_{-n+k}. \text{   for } k\geq 3 \label{neg_k_jacob}
\end{align} 
The next table gives the first terms of negatively subscripted order-$k$ Jacobsthal sequences for $n\geq 1$,
\begin{center}
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{|l|c|c|c|}
\hline
       & $k=2$ & $k=3$ & $k=4$  \\\hline
$i=1$  & $ \dfrac{\mathstrut 1}{\mathstrut 2}, -\dfrac{1}{4}, \dfrac{3}{8}, -\dfrac{5}{16}, \dfrac{11}{32}, -\dfrac{21}{64}, \dfrac{43}{128}, -\dfrac{85}{256} \ldots$ & $1, -2, 3, -3, 1, 4, -12, 21, -26, 19 \ldots$ & $1, -1, -1, 2, 2, -6, -1, 13, -3\ldots$\\ \hline
$i=2$ & $ -\dfrac{\mathstrut 1}{\mathstrut 2}, \dfrac{3}{4}, -\dfrac{5}{8}, \dfrac{11}{16}, -\dfrac{21}{32}, \dfrac{43}{64}, -\dfrac{85}{128}, \dfrac{171}{256} \ldots$ & $-1, 3, -5, 6, -4, -3, 16, -33, 47 \ldots$ & $-1, 2, 0, -3, 0, 8, -5, -14, 16, 25 \ldots$\\ \hline
$i=3$ & & $ -2, 3, -3, 1, 4, -12, 21, -26, 19, 9 \ldots $ &  $-2, 1, 4, -4, -7, 12, 10, -31, -8 \ldots$\\ \hline
$i=4$ & & & $ -1, -1, 2, 2, -6, -1, 13, -3, -28\ldots$\\ \hline
\end{tabular}
\end{adjustbox}
\captionof{table}{First terms of negatively subscripted order-$k$ Jacobsthal sequences}
\end{center}

\begin{remark}
The sequences of $k=3$ are known in the OEIS as \seqnum{A077990}, \seqnum{-A078064}, and \seqnum{A077990} respectively.
\end{remark}

We now give some results on the $n^{\text{th}}$ negatively subscripted generalized order-$k$ Fibonacci number ${g}^{(i)}_{-n}$ using matrix methods.
\begin{theorem} \label{theorem_term}
For $n\geq 0$ and $2\leq i\leq k$, we have
\begin{equation}
g^{(i)}_{-n-1}=-\frac{c_{i-1}}{c_{k}}g^{(k)}_{-n}+g^{(i-1)}_{-n}, \label{A}
\end{equation}
and for $i=1$, 
\begin{equation}
g^{(1)}_{-n-1}=\frac{1}{c_{k}}g^{(k)}_{-n}. \label{B}
\end{equation} 
\end{theorem} 

\begin{proof}
For the proof, we use the matrix approach. The matrix approach was used, for example, by Kalman \cite{kalman} and Er \cite{er} to study generalized Fibonacci numbers. Let
$$A:=
\begin{pmatrix}
-\frac{c_{k-1}}{c_{k}} & -\frac{c_{k-2}}{c_{k}} & -\frac{c_{k-3}}{c_{k}} & \cdots & -\frac{c_{1}}{c_{k}} & \frac{1}{c_{k}} \\ 
1 & 0 & 0 & \cdots & 0 & 0 \\ 
0 & 1 & 0 &  \cdots & 0 & 0 \\ 
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & 1 & 0 
\end{pmatrix}.$$
We have the following $n$ order-$k$ recurrence relations
\begin{equation}
\begin{pmatrix}
g^{(i)}_{-n-1} \\ g^{(i)}_{-n} \\ \vdots \\ g^{(i)}_{-n+k-2}
\end{pmatrix}=A\begin{pmatrix}
g^{(i)}_{-n} \\ g^{(i)}_{-n+1} \\ \vdots \\ g^{(i)}_{-n+k-1}
\end{pmatrix}. \label{vec_rec}
\end{equation}
In order to deal with $k$ sequences of negatively subscripted generalized order-$k$ Fibonacci sequences at the same time, we construct a $k \times k$ square matrix $G_{-n}$ as follows:
$$G_{-n}=
\begin{pmatrix}
g^{(k)}_{-n} & g^{(k-1)}_{-n} & g^{(k-2)}_{-n} & \cdots &  g^{(1)}_{-n}\\ 
g^{(k)}_{-n+1} & g^{(k-1)}_{-n+1} & g^{(k-2)}_{-n+1} & \cdots & g^{(1)}_{-n+1} \\ 
\vdots & \vdots & \vdots & \vdots &  \vdots \\
g^{(k)}_{-n+k-1} & g^{(k-1)}_{-n+k-1} & g^{(k-2)}_{-n+k-1} &  \cdots & g^{(1)}_{-n} 
\end{pmatrix}.$$
It is clear that, $A=G_{-1}$.

Next, \eqref{vec_rec} becomes
$$G_{-n-1}=AG_{-n}.$$
By induction, it is also clear that
$$G_{-n-1}=A^{n}A=G_{-n}A.$$ 
Finally, we have
\begin{equation}
G_{-n-1}=AG_{-n}=G_{-n}A. \label{C}
\end{equation}
Thus, Equations \eqref{A} and \eqref{B} are deduced from \eqref{C}.
\end{proof}


From Theorem \ref{theorem_term}, we get the following results:

\begin{corollary}
Let $f^{(i)}_{-n}$ be the $n^{\text{th}}$ negatively subscripted order-$k$ Fibonacci number of the sequence number $i$. Then
\begin{align*}
        f^{(1)}_{-n-1}&=f^{(k)}_{-n};\\
        f^{(i)}_{-n-1}&=-f^{(k)}_{-n}+f^{(i-1)}_{-n};  2\leq i\leq k.
\end{align*}
\end{corollary}

\begin{corollary}
Let $P^{(i)}_{-n}$ be the $n^{\text{th}}$ negatively subscripted order-$k$ Pell number of the sequence number $i$. Then
\begin{align*}
        P^{(1)}_{-n-1}&=P^{(k)}_{-n};\\
        P^{(2)}_{-n-1}&=-2P^{(k)}_{-n}+P^{(1)}_{-n};\\
        P^{(i)}_{-n-1}&=-P^{(k)}_{-n}+P^{(i-1)}_{-n};\ 3\leq i\leq k.
\end{align*}
\end{corollary}

\begin{corollary}
Let $J^{(i)}_{-n}$ be the $n^{\text{th}}$ negatively subscripted generalized order-$k$ Jacobsthal number of the sequence number $i$. Then for $k\geq 3$,
\begin{align*}
        J^{(1)}_{-n-1}&=J^{(k)}_{-n};\\
        J^{(2)}_{-n-1}&=-J^{(k)}_{-n}+J^{(1)}_{-n};\\
        J^{(3)}_{-n-1}&=-2J^{(k)}_{-n}+J^{(2)}_{-n};\\
        J^{(i)}_{-n-1}&=-J^{(k)}_{-n}+J^{(i-1)}_{-n};\ 4\leq i\leq k.
\end{align*}
\end{corollary}

We conclude this section by defining a generalized order-$k$ Fibonacci sequence $(V_{n})_{n}$ with arbitrary initial conditions and we give the extension of the sequence $(V_{n})_{n}$ to negative indices. Then we give the $n^{\text{th}}$ negatively subscripted number $V_{-n}$ as the sum of $k$ terms of negatively subscripted generalized order-$k$ Fibonacci numbers defined in \eqref{neg_k_fibo}. 

For $n\geq k$, let $V_{n}$ be as follows:
\begin{equation*}
V_{n}= c_{1}V_{n-1}+c_{2}V_{n-2}+\cdots+c_{k}V_{n-k}, \label{k_V}
\end{equation*}
with integer initial conditions $V_{0},V_{1},\dots, V_{k-1}$, where $c_{1},c_{2},\dots,c_{k}\in \mathbb{R}$.

Belbachir and Bencherif \cite{belbachir} gave the explicit formula of the $n^{\text{th}}$ term of the sequence $(V_{n})_{n}$.  

Then for $n\geq 1$, the value $V_{-n}$ is as follows:
\begin{equation}
V_{-n}= -\frac{c_{k-1}}{c_{k}}V_{-n+1}-\frac{c_{k-2}}{c_{k}}V_{-n+2}-\cdots-\frac{c_{1}}{c_{k}}V_{-n+k-1}+\frac{1}{c_{k}}V_{-n+k}. \label{neg_V}
\end{equation}

Note that for $(V_{0},\dots,V_{k-i-1},V_{k-i},V_{k-i+1},\dots,V_{k-1})=(0,\dots,0,1,0,\dots,0)$, $1\leq i \leq k$, we have $V_{n}=g^{(i)}_{n}$.

The $n^{\text{th}}$ term $V_{-n}$ is given by
\begin{theorem}
For $n\geq 1$,
\begin{equation}
V_{-n}=\sum_{i=0}^{k-1}a_{i}g_{-n}^{(k-i)}, \label{rec}
\end{equation}
where $g^{(j)}_{-n}$ is the $n^{\text{th}}$ negatively subscripted generalized order-$k$ Fibonacci number of the sequence number $i$ \eqref{neg_k_fibo}.
\end{theorem}
\begin{proof}
By induction on $n$.
 
For $n=1$, $V_{-1}= -\frac{c_{k-1}}{c_{k}}V_{0}-\cdots-\frac{c_{1}}{c_{k}}V_{k-2}+\frac{1}{c_{k}}V_{k-1}=-\frac{c_{k-1}}{c_{k}}a_{0}-\cdots-\frac{c_{1}}{c_{k}}a_{k-2}+\frac{1}{c_{k}}a_{k-1}=\sum_{i=0}^{k-1}a_{i}g_{-1}^{(k-i)}$. 

Now suppose that \eqref{rec} is true for $2\leq j \leq n$ and we prove the formula for $n+1$:
\begin{align*}
        V_{-n-1}&=-\frac{c_{k-1}}{c_{k}}V_{-n}-\frac{c_{k-2}}{c_{k}}V_{-n+1}-\cdots-\frac{c_{1}}{c_{k}}V_{-n+k-2}+\frac{1}{c_{k}}V_{-n+k-1}.
\end{align*}
and
\begin{align*}
        V_{-n-1}&=-\frac{c_{k-1}}{c_{k}}\sum_{i=0}^{k-1}a_{i}g_{-n}^{(k-i)}-\frac{c_{k-2}}{c_{k}}\sum_{i=0}^{k-1}a_{i}g_{-n+1}^{(k-i)}-\cdots-\frac{c_{1}}{c_{k}}\sum_{i=0}^{k-1}a_{i}g_{-n+k-2}^{(k-i)}
        +\frac{1}{c_{k}}\sum_{i=0}^{k-1}a_{i}g_{-n+k-1}^{(k-i)}\\
&=a_{0}\left( -\frac{c_{k-1}}{c_{k}}g_{-n}^{(k)}-\frac{c_{k-2}}{c_{k}}g_{-n+1}^{(k)}-\cdots-\frac{c_{1}}{c_{k}}g_{-n+k-2}^{(k)}+\frac{1}{c_{k}}g_{-n+k-1}^{(k)} \right)\\
&+ a_{1}\left( -\frac{c_{k-1}}{c_{k}}g_{-n}^{(k-1)}-\frac{c_{k-2}}{c_{k}}g_{-n+1}^{(k-1)}-\cdots-\frac{c_{1}}{c_{k-1}}g_{-n+k-2}^{(k-1)}+\frac{1}{c_{k-1}}g_{-n+k-1}^{(k-1)} \right)\\
& + \cdots + \\ 
&+a_{k-1}\left( -\frac{c_{k-1}}{c_{k}}g_{-n}^{(1)}-\frac{c_{k-2}}{c_{k}}g_{-n+1}^{(1)}-\cdots-\frac{c_{1}}{c_{k}}g_{-n+k-2}^{(1)}+\frac{1}{c_{k}}g_{-n+k-1}^{(1)}\right)\\
&=a_{0}g_{-n-1}^{(k)}+ a_{1}g_{-n-1}^{(k-1)}+\cdots+a_{k-1}g_{-n-1}^{(1)}\\
&=\sum_{i=0}^{k-1}a_{i}g_{-n-1}^{(k-i)}.
\end{align*}
Thus \eqref{rec} is true for all $n\geq 1$.
\end{proof}

\section{Representation of negatively subscripted generalized order-$k$ Fibonacci numbers}\label{representations}

In this section, we give a representation of the family of negatively subscripted generalized order-$k$ Fibonacci numbers \eqref{neg_k_fibo} using the permanent of Hessenberg matrices. 

We introduce the $n\times n$ matrix $W_{n}=(w_{st})$ with $w_{1t}=0$, for $1\leq t \leq n$ and for all $1<i\leq n$,
\begin{displaymath}
w_{st}=\begin{cases}
1, & \text{if $s=t+1$;} \\ 
-\frac{c_{k-(t-s)-1}}{c_{k}}, & \text{if $0\leq t-s\leq k-2$;} \\ 
\frac{1}{c_{k}}, & \text{if $s=t-k+1$;} \\ 
0, & \text{elsewhere.}
\end{cases}
\end{displaymath}
And we construct, for $n\geq 1$, the $n\times n$ matrix $A_{n}^{i}$, for $1\leq i\leq k$, as follows:
\begin{equation*}
A_{n}^{i}=W_{n}+\sum_{t=1}^{n}\left(\frac{\delta_{i,k}^{t}(k)}{c_{k}}-\sum_{j=1}^{k-1}\delta_{i,j}^{t}(k)\frac{c_{k-j}}{c_{k}}\right)E_{1t}
\end{equation*}
with $\delta_{a,b}^{c}(d)$ defined for $1\leq a,b,c\leq d$ as 
\begin{equation}
\delta_{a,b}^{c}(d) = \begin{cases}
        1, & \text{if $a+b = c+d$;} \\
        0, & \text{otherwise.}
    		\end{cases}
\label{kronecker}
\end{equation}
And $E_{st}$ denotes the $n\times n$ matrix with $1$ in position $(s,t)$ and $0$ elsewhere.

Then $A_{n}^{i}$ is as follows:
\begin{equation}
A_{n}^{i}=\begin{pmatrix}
A_{1} & A_{2} & A_{3} & \cdots & A_{k} & 0 & 0 & \cdots & 0 \\ 
1 & -\dfrac{c_{k-1}}{c_{k}} & -\dfrac{c_{k-2}}{c_{k}} & \cdots & -\dfrac{c_{1}}{c_{k}} & \dfrac{1}{c_{k}} & 0 & \cdots & 0 \\ 
0 & 1 & -\dfrac{c_{k-1}}{c_{k}} &  \cdots & -\dfrac{c_{2}}{c_{k}} & -\dfrac{c_{1}}{c_{k}} & \dfrac{1}{c_{k}} & \ddots & \vdots \\ 
&  & & &  &  &  & \ddots & 0 \\ 
\vdots &  & & &  &  &  & \ddots & \dfrac{1}{c_{k}} \\ 
&  & & &  &  &  & \ddots & -\dfrac{c_{1}}{c_{k}} \\ 
 & & &  &  &  &  & \ddots & \vdots \\ 
0 & \cdots & &  &  & \cdots & 0 & 1 & -\dfrac{c_{k-1}}{c_{k}} 
\end{pmatrix},\label{mat_rep}
\end{equation}
where $A_{t}=\frac{\delta_{i,k}^{t}(k)}{c_{k}}-\sum_{j=1}^{k-1}\delta_{i,j}^{t}(k)\frac{c_{k-j}}{c_{k}}$, for $1\leq t \leq k$.

We now give the representation of negatively subscripted generalized order-$k$ Fibonacci numbers.
\begin{theorem}
For $n\geq 1$, let $A_{n}^{i}$ be as in \eqref{mat_rep} and $g_{-n}^{(i)}$ be the $n^{\text{th}}$ negatively subscripted generalized order-$k$ Fibonacci number of the sequence number $i$, then
\begin{equation}
\per A_{n}^{i}= g_{-n}^{(i)}.
\end{equation}
\label{theorem_representation}
\end{theorem}
For instance, we give the following representations of $k$ sequences of negatively subscripted order-$k$ Fibonacci, order-$k$ Pell, and order-$k$ Jacobsthal numbers.

Let $F_{n}^{i}$ be the $n\times n$ matrix defined, for $1\leq i\leq k$, as follows:
\begin{multline}
F_{n}^{i}=
\resizebox{.8\textwidth}{!}{$
\begin{pmatrix}
\delta_{i,k}^{1}(k)-\sum_{j=1}^{k-1}\delta_{i,j}^{1}(k) & \delta_{i,k}^{2}(k)-\sum_{j=1}^{k-1}\delta_{i,j}^{2}(k) & \delta_{i,k}^{3}(k)-\sum_{j=1}^{k-1}\delta_{i,j}^{3}(k) & \cdots & \delta_{i,k}^{k}-\sum_{j=1}^{k-1}\delta_{i,j}^{k}(k) & 0 & 0 & \cdots & 0 \\ 
1 & -1 & -1 & \cdots & -1 & 1 & 0 & \cdots & 0 \\ 
0 & 1 & -1 &  \cdots & -1 & -1 & 1 & \ddots & \vdots \\ 
&  & & &  &  &  & \ddots & 0 \\ 
\vdots &  & & &  &  &  & \ddots & 1 \\ 
&  & & &  &  &  & \ddots & -1 \\ 
 & & &  &  &  &  & \ddots & \vdots \\ 
0 & \cdots & &  &  & \cdots & 0 & 1 & -1
\end{pmatrix}.\label{mat_fibo} $}
\end{multline}
where $\delta_{i,k}^{t}(k)$ and $\delta_{i,j}^{t}(k)$ is defined as in \eqref{kronecker} for all $1 \leq t \leq k$ and $1 \leq j \leq k-1$.

\begin{corollary}
For $n\geq 1$, let $F_{n}^{i}$ be as in \eqref{mat_fibo} and $f_{-n}^{(i)}$ be the $n^{\text{th}}$ negatively subscripted order-$k$ Fibonacci number of the sequence number $i$, then
\begin{equation}
\per F_{n}^{i}= f_{-n}^{(i)}.
\end{equation}
\end{corollary}

Let $\mathcal{P}_{n}^{i}$ be the $n\times n$ matrix defined for $1\leq i\leq k$ in the following way:

\begin{equation}
\mathcal{P}_{n}^{i}=
\begin{pmatrix}
S_{1} & S_{2} & \cdots & S_{k-1} & S_{k} & 0 & 0 & \cdots & 0 \\ 
1 & -1 & \cdots & -1 & -2 & 1 & 0 & \cdots & 0 \\ 
0 & 1  & -1 & \cdots & -1 & -2 & 1 & \ddots & \vdots  \\ 
&  & & &  &  &  & \ddots & 0 \\ 
\vdots &  & & &  &  &  & \ddots & 1 \\ 
&  & & &  &  &  & \ddots & -2 \\ 
 & & &  &  &  &  & \ddots & \vdots \\ 
0 & \cdots & &  &  & \cdots & 0 & 1 & -1
\end{pmatrix}, \label{mat_pell}
\end{equation}
where $S_{t}=\delta_{i,k}^{t}-2\delta_{i,k-1}^{t}-\sum_{j=1}^{k-2}\delta_{i,j}^{t}$ for $1\leq t \leq k$ and $\delta_{i,j}^{t}(k)$ is defined as in \eqref{kronecker}.

\begin{corollary}
For $n\geq 1$, let $\mathcal{P}_{n}^{i}$ be as in \eqref{mat_pell} and $P_{-n}^{(i)}$ be the $n^{\text{th}}$ negatively subscripted order-$k$ Pell number of the sequence number $i$, then
\begin{equation}
\per \mathcal{P}_{n}^{i}= P_{-n}^{(i)}.
\end{equation}
\end{corollary}

Let $\mathcal{J}_{n}^{i}$ be the $n\times n$ matrix defined for $1\leq i\leq k$ and $k\geq 3$ as follows:

\begin{equation}
\mathcal{J}_{n}^{i}=
\begin{pmatrix}
T_{1} & T_{2} & T_{3} & \cdots & T_{k-2} & T_{k-1} & T_{k} & 0 & 0 & \cdots & 0 \\ 
1 & -1 & -1 & \cdots & -1 & -2 & -1 & 1 & 0 & \cdots & 0 \\ 
0 & 1 & -1 &  \cdots & -1 & -1 & -2 & -1 & 1 & \ddots & \vdots \\ 
& & & & & & &  &  & \ddots & 0 \\ 
\vdots & & & & & &  &  &  & \ddots & 1 \\ 
&  & & & & & &  &  & \ddots & -1 \\ 
 & & & & & &  &  &  & \ddots & -2 \\ 
  & & & & & &  &  &  & \ddots & -1 \\
 & & & & & &  &  &  & \ddots & \vdots \\ 
0 & \cdots & & & &  &  & \cdots & 0 & 1 & -1
\end{pmatrix}, \label{mat_jacob}
\end{equation}
where $T_{t}=\delta_{i,k}^{t}-2\delta_{i,k-2}^{t}-\delta_{i,k-1}^{t}-\sum_{j=1}^{k-3}\delta_{i,j}^{t}$ for $1\leq t \leq k$ and $\delta_{i,j}^{t}(k)$ is defined as in \eqref{kronecker}.

\begin{corollary}
For $n\geq 1$, let $\mathcal{J}_{n}^{i}$ be as in \eqref{mat_jacob} and $J_{-n}^{(i)}$ be the $n^{\text{th}}$ negatively subscripted order-$k$ Jacobsthal number of the sequence number $i$, then
\begin{equation}
\per \mathcal{J}_{n}^{i}= J_{-n}^{(i)}.
\end{equation}
\end{corollary}

\section{More permanental representations of negatively subscripted order-$k$ Fibonacci and order-$k$ Lucas numbers}\label{more representations}

In this section, we give numerous types of matrices whose permanent are negatively subscripted order-$k$ Fibonacci and order-$k$ Lucas terms.

We first establish two kinds of permanental representations of the negatively subscripted order-$k$ Fibonacci sequence $(\widetilde{g}_{-n})_{n}$. We introduce an $n\times n$ $(0,1,-1)$-matrix $B_{(n,k)}$ defined by
\begin{equation}
b_{ij}=\begin{cases}
1, & \text{for $j-i=-1$ or $k-1$;} \\ 
-1, & \text{for $0\leq j-i\leq k-2$;} \\ 
0, & \text{elsewhere.}
\end{cases}
\label{matrix_B}
\tag{$\star$}
\end{equation}
And we construct for $n>k$, the $n\times n$ $(0,1,-1)$-matrices $F_{(n,k)}$ and $\widetilde{F}_{(n,k)}$
as follows:
\begin{equation*}
F_{(n,k)}=B_{(n,k)}+2E_{11}+\overset{k-1}{\underset
{j=2}{\sum }}E_{1j}-E_{1k},
\end{equation*}
\begin{equation*}
\widetilde{F}_{(n,k)}=B_{(n,k)}+2\sum\nolimits_{j=1}^{k-1}E_{1j},
\end{equation*}
where $E_{ij}$ denotes the $n\times n$ matrix with $1$ in position $(i,j)$ and $0$ elsewhere. $F_{(n,k)}$ and $\widetilde{F}_{(n,k)}$ correspond to
\begin{equation*}
F_{(n,k)}=
\begin{pmatrix}
1 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 \\ 
1 & -1 & \cdots & -1 & 1 & 0 & \cdots & 0 \\ 
& \ddots & \ddots &  & \ddots & \ddots & \ddots & \vdots \\ 
&  &  &  &  &  & \ddots & 0 \\ 
&  &  &  &  &  & \ddots & 1 \\ 
&  &  &  &  &  &  & -1 \\ 
\vdots &  &  &  &  & \ddots & \ddots & \vdots \\ 
0 &  &  &  &  &  & 1 & -1
\end{pmatrix}
\end{equation*}
and
\begin{equation*}
\widetilde{F}_{(n,k)}=
\begin{pmatrix}
1 & 1 & \cdots & 1 & 0 & 0 & \cdots & 0 \\ 
1 & -1 & \cdots & -1 & 1 & 0 & \cdots & 0 \\ 
& \ddots & \ddots &  & \ddots & \ddots & \ddots & \vdots \\ 
&  &  &  &  &  & \ddots & 0 \\ 
&  &  &  &  &  & \ddots & 1 \\ 
&  &  &  &  &  &  & -1 \\ 
\vdots &  &  &  &  & \ddots & \ddots & \vdots \\ 
0 &  &  &  &  &  & 1 & -1
\end{pmatrix}.
\end{equation*}

Then we give the two permanental representations of the negatively subscripted order-$k$ Fibonacci sequence.
\begin{theorem}
\label{theorem-rep-k-fibo}For $n>k$, we have
\begin{equation}
\per F_{(n,k)}=\widetilde{g}_{-n}^{(k)}  \label{5.2}
\end{equation}
and
\begin{equation}
\per \widetilde{F}_{(n,k)}=\widetilde{g}_{-n+k}^{(k)}.
\label{5.3}
\end{equation}
\end{theorem}

Next, using the same approach given in Theorem \ref{theorem-rep-k-fibo}, see next section, we give a representation of $l_{-n+1}^{(k)}$.

We use the $n\times n$ square matrix $B_{(n,k)}=(b_{ij})$ given by \eqref{matrix_B}, and we construct for $n>k$, the $n\times n$ matrix $A_{(n,k)}$ as follows:
\begin{equation*}
A_{(n,k)}=B_{(n,k)}+2E_{11}+\overset{k-1}{\underset
{j=2}{\sum }}E_{1j}-E_{1k}+E_{1,k+1},
\end{equation*}
where $E_{ij}$ is the $n\times n$ matrix with $1$ in position $(i,j)$ and $0$ elsewhere.

The matrix $A_{(n,k)}$ corresponds to
\begin{equation*}
A_{(n,k)}=
\begin{pmatrix}
1 & 0 & \cdots  & 0 & 0 & 1 & 0 & \cdots  & 0 \\ 
1 & -1 & \cdots  & -1 & -1 & 1 & 0 & \ddots  & \vdots  \\ 
& \ddots  & \ddots  &  & \ddots  & -1 &  &  & 0 \\ 
&  &  & \ddots  &  & \ddots  & \ddots  & \ddots  & \vdots  \\ 
&  &  &  &  &  &  &  & 1 \\ 
&  &  &  &  &  &  &  & -1 \\ 
&  &  &  &  &  &  & \ddots  & \vdots  \\ 
&  &  &  &  &  & \ddots  & \ddots  & \vdots  \\ 
0 &  & \cdots  &  &  & 0 &  & 1 & -1
\end{pmatrix}.
\end{equation*}

\begin{theorem}
\label{theorem-rep-k-lucas}For $n>k$, we have
\begin{equation*}
\per A_{(n,k)}=l_{-n+1}^{(k)}.
\end{equation*}
\end{theorem}
\begin{proof}
The proof is the same as Theorem \ref{theorem-rep-k-fibo}, see next section.
\end{proof}
Finally, let $B_{n}$ be an $n\times n$ matrix defined by
\begin{equation*}
B_{n}=T_{n}+2(E_{11}+E_{22}+E_{33}-E_{43})+E_{13}-E_{23}+E_{24}-E_{34},
\end{equation*}
where $T_{n}$ is the $n\times n$ $(0,1,-1)$-matrix
such that $t_{ii}=-1$, $t_{ij}=1$ if and only if $\left\vert j-i\right\vert
=1$.

The matrix $B_{n}$ corresponds to
\begin{equation*}
B_{n}=
\begin{pmatrix}
1 & 1 & 1 & 0 & 0 & \cdots &  & 0 \\ 
1 & 1 & 0 & 1 & 0 &  &  & \vdots \\ 
0 & 1 & 1 & 0 & 0 &  &  &  \\ 
0 & 0 & -1 & -1 & 1 & 0 &  &  \\ 
0 & 0 & 0 & 1 & -1 & 1 & \ddots & \vdots \\ 
\vdots &  &  &  & \ddots & \ddots & \ddots & 0 \\ 
&  &  &  & \ddots & \ddots & \ddots & 1 \\ 
0 & \cdots &  &  &  & 0 & 1 & -1
\end{pmatrix}.
\end{equation*}

Then we obtain the following result.
\begin{theorem}
\label{theorem-second-lucas}For $n\geq 4$, we have
\begin{equation*}
\per B_{n}=l_{-n+1}^{(2)}.
\end{equation*}
\end{theorem}

\section{Proofs}\label{proofs}

We start this section by giving the proof of Theorem \ref{theorem_representation},
\begin{proof}
We have
$$\frac{\delta_{i,k}^{1}}{c_{k}}-\sum_{t=1}^{k-1}\delta_{i,t}^{1}\frac{c_{k-t}}{c_{k}} = \begin{cases}
        \frac{1}{c_{k}}, & \text{for $i=1$;} \\
        -\frac{c_{i-1}}{c_{k}}, & \text{for $2\leq i \leq k$.}
    \end{cases}$$

On the other hand by Theorem \ref{theorem_term}, we have
$$g_{-1}^{(i)} = \begin{cases}
        \frac{1}{c_{k}}, & \text{for $i=1$;} \\
        -\frac{c_{i-1}}{c_{k}}, & \text{for $2\leq i \leq k$.}
    \end{cases}$$
  
Thus, 
$$\frac{\delta_{i,k}^{1}}{c_{k}}-\sum_{t=1}^{k-1}\delta_{i,t}^{1}\frac{c_{k-t}}{c_{k}}=g_{-1}^{(i)}.$$

Next for $2\leq j \leq k-1$, by the definition of shifted generalized order-$k$ Fibonacci sequences \eqref{k_fibo}, we have
$$\frac{\delta_{i,k}^{j}}{c_{k}}-\sum_{t=1}^{k-1}\delta_{i,t}^{j}\frac{c_{k-t}}{c_{k}} = \begin{cases}
    	0, & \text{for $1\leq i \leq j-1$;} \\
        \frac{1}{c_{k}}g^{(i)}_{k-j}, & \text{for $i=j$;} \\
        -\frac{c_{i-1}}{c_{k}}g^{(i)}_{k-i}, & \text{for $j+1\leq i \leq k$.}
    \end{cases}$$

Then $\frac{\delta_{i,k}^{j}}{c_{k}}-\sum_{t=1}^{k-1}\delta_{i,t}^{j}\frac{c_{k-t}}{c_{k}}$ can be written as
$$\frac{\delta_{i,k}^{j}}{c_{k}}-\sum_{t=1}^{k-1}\delta_{i,t}^{j}\frac{c_{k-t}}{c_{k}}=\frac{1}{c_{k}}g^{(i)}_{k-j}-\sum_{t=j}^{k-1}\frac{c_{k-t}}{c_{k}}g_{t-j}^{(i)}.$$

Finally, 
$$\frac{\delta_{i,k}^{k}}{c_{k}}-\sum_{t=1}^{k-1}\delta_{i,t}^{k}\frac{c_{k-t}}{c_{k}} = \begin{cases}
        \frac{1}{c_{k}}g_{0}^{(k)}, & \text{for $i=k$;} \\
        0, & \text{otherwise.}
    \end{cases}$$

Then for $1\leq i \leq k$, we have $$\frac{\delta_{i,k}^{k}}{c_{k}}-\sum_{t=1}^{k-1}\delta_{i,t}^{k}\frac{c_{k-t}}{c_{k}}=\frac{1}{c_{k}}g_{0}^{(i)}.$$
Hence, the matrix $A_{n}^{i}$ can be written as follows:

\begin{displaymath}
A_{n}^{i}=
\resizebox{.8\textwidth}{!}{$
\begin{pmatrix}
g_{-1}^{(i)} \hspace{.1cm} & \dfrac{1}{c_{k}}g^{(i)}_{k-2}-\sum_{t=2}^{k-1}\dfrac{c_{k-t}}{c_{k}}g_{t-2}^{(i)} \hspace{.1cm} & \dfrac{1}{c_{k}}g^{(i)}_{k-3}-\sum_{t=3}^{k-1}\dfrac{c_{k-t}}{c_{k}}g_{t-3}^{(i)} \hspace{.1cm} & \cdots & \dfrac{1}{c_{k}}g_{0}^{(i)} & 0 & 0 & \cdots & 0 \\ 
1 & -\dfrac{c_{k-1}}{c_{k}} & -\dfrac{c_{k-2}}{c_{k}} & \cdots & -\dfrac{c_{1}}{c_{k}} & \dfrac{1}{c_{k}} & 0 & \cdots & 0 \\ 
0 & 1 & -\dfrac{c_{k-1}}{c_{k}} &  \cdots & -\dfrac{c_{2}}{c_{k}} & -\dfrac{c_{1}}{c_{k}} & \dfrac{1}{c_{k}} & \ddots & \vdots \\ 
&  & & &  &  &  & \ddots & 0 \\ 
\vdots &  & & &  &  &  & \ddots & \dfrac{1}{c_{k}} \\ 
&  & & &  &  &  & \ddots & -\dfrac{c_{1}}{c_{k}} \\ 
 & & &  &  &  &  & \ddots & \vdots \\ 
0 & \cdots & &  &  & \cdots & 0 & 1 & -\dfrac{c_{k-1}}{c_{k}}
\end{pmatrix}.$}
\end{displaymath}

By Lemma \ref{Brualdi}, the matrix $A_{n}^{i}$ is contractible on column $1$ relative to rows $1$ and $2$. Let $(A_{n}^{i})^{1}$ be the $(n-1) \times (n-1)$ contraction matrix, then we have $\per A_{n}^{i}= \per (A_{n}^{i})^{1}$ and $(A_{n}^{i})^{1}$ is as follows:
\begin{displaymath}
(A_{n}^{i})^{1}=
\resizebox{.8\textwidth}{!}{$
\begin{pmatrix}
g_{-2}^{(i)} \hspace{.1cm} & \dfrac{1}{c_{k}}g^{(i)}_{k-3}-\sum_{t=2}^{k-1}\dfrac{c_{k-t}}{c_{k}}g_{t-3}^{(i)} \hspace{.1cm} & \dfrac{1}{c_{k}}g^{(i)}_{k-4}-\sum_{t=3}^{k-1}\dfrac{c_{k-t}}{c_{k}}g_{t-4}^{(i)} \hspace{.1cm} & \cdots & \dfrac{1}{c_{k}}g_{-1}^{(i)} & 0 & 0 & \cdots & 0 \\ 
1 & -\dfrac{c_{k-1}}{c_{k}} & -\dfrac{c_{k-2}}{c_{k}} & \cdots & -\dfrac{c_{1}}{c_{k}} & \dfrac{1}{c_{k}} & 0 & \cdots & 0 \\ 
0 & 1 & -\dfrac{c_{k-1}}{c_{k}} &  \cdots & -\dfrac{c_{2}}{c_{k}} & -\dfrac{c_{1}}{c_{k}} & \dfrac{1}{c_{k}} & \ddots & \vdots \\ 
&  & & &  &  &  & \ddots & 0 \\ 
\vdots &  & & &  &  &  & \ddots & \dfrac{1}{c_{k}} \\ 
&  & & &  &  &  & \ddots & -\dfrac{c_{1}}{c_{k}} \\ 
 & & &  &  &  &  & \ddots & \vdots \\ 
0 & \cdots & &  &  & \cdots & 0 & 1 & -\dfrac{c_{k-1}}{c_{k}}
\end{pmatrix}.$}
\end{displaymath}

Furthermore, the matrix $(A_{n}^{i})^{1}$ is also contractible on column $1$ relative to rows $1$ and $2$. Continuing the same contraction process we get by Lemma \ref{Brualdi}, 
$$\per A_{n}^{i}= \per (A_{n}^{i})^{p}$$
with $(A_{n}^{i})^{p}$ defined for $p\leq n-k$ by
\begin{displaymath}
(A_{n}^{i})^{p}=
\resizebox{.8\textwidth}{!}{$
\begin{pmatrix}
g_{-p-1}^{(i)} \hspace{.1cm} & \dfrac{1}{c_{k}}g_{-p+k-2}^{(i)}-\sum_{t=2}^{k-1}\dfrac{c_{k-t}}{c_{k}}g_{-p+t-2}^{(i)} & \cdots & \dfrac{1}{c_{k}}g_{-p+1}^{(i)}-\dfrac{c_{1}}{c_{k}}g_{-p}^{(i)} & \dfrac{1}{c_{k}}g_{-p}^{(i)} & 0 & 0 & \cdots & 0 \\ 
1 & -\dfrac{c_{k-1}}{c_{k}}  & \cdots & -\dfrac{c_{2}}{c_{k}} & -\dfrac{c_{1}}{c_{k}} & \dfrac{1}{c_{k}} & 0 & \cdots & 0 \\ 
0 & 1  &  \cdots & -\dfrac{c_{3}}{c_{k}} & -\dfrac{c_{2}}{c_{k}} & -\dfrac{c_{1}}{c_{k}} & \dfrac{1}{c_{k}} & \ddots & \vdots \\ 
&  &  &  &  & & & \ddots & 0 \\ 
\vdots &   & &  &  & & & \ddots & \dfrac{1}{c_{k}} \\ 
&   & &  &  & & & \ddots & -\dfrac{c_{1}}{c_{k}} \\ 
 &  &  &  &  & & & \ddots & \vdots \\ 
0 & \cdots  &  &  & \cdots & 0 & 0 & 1 & -\dfrac{c_{k-1}}{c_{k}}
\end{pmatrix}.$}
\end{displaymath}
And for $p\geq n-k+1$, $(A_{n}^{i})^{p}$ is defined as follows:

\begin{displaymath}
(A_{n}^{i})^{p}=
\resizebox{.8\textwidth}{!}{$
\begin{pmatrix}
g_{-p-1}^{(i)} & \frac{1}{c_{k}}g_{-p+k-2}^{(i)}-\sum_{t=2}^{k-1}\frac{c_{k-t}}{c_{k}}g_{-p+t-2}^{(i)} & \cdots & \frac{1}{c_{k}}g_{-n+k+1}^{(i)}-\sum_{t=n-p+1}^{k-1}\frac{c_{k-t}}{c_{k}}g_{-n+t+1}^{(i)} & \frac{1}{c_{k}}g_{-n+k}^{(i)}-\sum_{t=n-p}^{k-1}\frac{c_{k-t}}{c_{k}}g_{-n+t}^{(i)} \\ 
1 & -\dfrac{c_{k-1}}{c_{k}}  & \cdots & -\dfrac{c_{k-n+p+2}}{c_{k}} & -\dfrac{c_{k-n+p+1}}{c_{k}} \\ 
0 & 1  &  \cdots & -\dfrac{c_{k-n+p+3}}{c_{k}} & -\dfrac{c_{k-n+p+2}}{c_{k}}  \\ 
\vdots &  &  &  &  \vdots \\ 
0 & \cdots & 0 & 1 & -\dfrac{c_{k-1}}{c_{k}}
\end{pmatrix}.$}
\end{displaymath}
Then 
$$(A_{n}^{i})^{n-2}=
\begin{pmatrix}
g_{-n+1}^{(i)} & \dfrac{1}{c_{k}}g_{-n+k}^{(i)}-\sum_{t=2}^{k-1}\dfrac{c_{k-t}}{c_{k}}g_{-n+t}^{(i)} \\ 
1 & -\dfrac{c_{k-1}}{c_{k}}  \\
\end{pmatrix}.
$$
By Lemma \ref{Brualdi}, we have
\begin{align*}
        \per A_{n}^{i}= \per (A_{n}^{i})^{n-2} & =-\dfrac{c_{k-1}}{c_{k}}g_{-n+1}^{(i)}+\dfrac{1}{c_{k}}g_{-n+k}^{(i)}-\sum_{t=2}^{k-1}\dfrac{c_{k-t}}{c_{k}}g_{-n+t}^{(i)}  \\
        & =\dfrac{1}{c_{k}}g_{-n+k}^{(i)}-\sum_{t=1}^{k-1}\dfrac{c_{k-t}}{c_{k}}g_{-n+t}^{(i)} \\
        & =g_{-n}^{(i)}.
\end{align*}
\noindent So $\per A_{n}^{i}= g_{-n}^{(i)}$ and the proof is complete.
\end{proof}

Secondly, we provide the proof of Theorem \ref{theorem-rep-k-fibo},
\begin{proof}
Let $F_{(n,k)}^{p}=(f_{ij}^{p})$ be the $p^{\text{th}}$
contraction of $F_{(n,k)}$, $1\leq p\leq n-2$.

The matrix $F_{(n,k)}=(f_{ij})$, can be contracted on column $1$ relative to rows $1$ and $2$.

We can easily verify that if $p=1$,

$f_{11}^{1}=f_{12}^{1}=\cdots =f_{1k-1}^{1}=-1$, $f_{1k}^{1}=1$ and $
f_{1q}^{1}=0$, for $q\geq k+1$; and for all $i=2,\dots ,n-1$,
$$\begin{cases}
f_{i,i-1}=1, & \\ 
f_{ij}=-1,  & \text{for $0\leq j-i\leq k-2$;}\\ 
f_{ij}=1 & \text{for $j-i=k-1$.}
\end{cases} $$
Hence
\begin{equation*}
F_{(n,k)}^{1}=
\begin{pmatrix}
-1 & \cdots & -1 & 1 & 0 & \cdots & 0 \\ 
1 & -1 & \cdots & -1 & 1 & \ddots & \vdots \\ 
&  &  &  & \ddots & \ddots & 0 \\ 
&  &  &  &  & \ddots & 1 \\ 
\vdots &  &  &  &  &  & -1 \\ 
&  &  &  & \ddots & \ddots & \vdots \\ 
0 &  & \cdots &  &  & 1 & -1
\end{pmatrix}
\end{equation*}
and
\begin{displaymath}
F_{(n,k)}^{1}=
\resizebox{.8\textwidth}{!}{$
\begin{pmatrix}
\widetilde{g}_{-2}^{(k)} & \widetilde{g}_{k-3}^{(k)}-\sum\nolimits_{j=-1}^{k-4}\widetilde{g}_{j}^{(k)} & \widetilde{g}_{k-4}^{(k)}-\sum\nolimits_{j=-1}^{k-5}\widetilde{g}_{j}^{(k)} & \cdots & 
-\widetilde{g}_{-1}^{(k)}+\widetilde{g}_{0}^{(k)} & \widetilde{g}_{-1}^{(k)} & 0 & \cdots & 0 \\ 
1 & -1 & -1 & \cdots &  & -1 & 1 & \ddots & \vdots \\ 
0 & 1 & -1 & \cdots &  &  & -1 & \ddots & 0 \\ 
&  &  &  &  &  &  & \ddots & 1 \\ 
&  &  &  &  &  &  &  & -1 \\ 
&  &  &  &  &  & \ddots & \ddots & \vdots \\ 
0 &  &  & \cdots &  &  & 0 & 1 & -1
\end{pmatrix}.$}
\end{displaymath}

Furthermore, the matrix $F_{(n,k)}^{1}$ can be contracted.
From Lemma \ref{Brualdi} we obtain
$$\per F_{(n,k)}=\per F_{(n,k)}^{p}$$
with $F_{(n,k)}^{p}$ defined for $p\leq n-k$ by
\begin{equation*}
F_{(n,k)}^{p}=
\begin{pmatrix}
\widetilde{g}_{-p-1}^{(k)} & \widetilde{g}_{-p+k-2}^{(k)}-\sum\nolimits_{j=0}^{k-3}\widetilde{g}_{-p+j}^{(k)} & \cdots & 
-\widetilde{g}_{-p}^{(k)}+\widetilde{g}_{-p+1}^{(k)} & \widetilde{g}_{-p}^{(k)} & 0 & \cdots & 0 \\ 
1 & -1 & \cdots & \cdots & -1 & 1 & \ddots & \vdots \\ 
&  &  &  &  & \ddots & \ddots & 0 \\ 
&  &  &  &  &  & \ddots & 1 \\ 
&  &  &  &  &  &  & -1 \\ 
&  &  &  &  & \ddots & \ddots & \vdots \\ 
0 &  & \cdots &  &  & 0 & 1 & -1
\end{pmatrix}.
\end{equation*}

And for $p\geq n-k+1$, $F_{(n,k)}^{p}$ is defined as
\begin{displaymath}
F_{(n,k)}^{p}=
\resizebox{.8\textwidth}{!}{$
\begin{pmatrix}
\widetilde{g}_{-p-1}^{(k)} & \widetilde{g}_{-p+k-2}^{(k)}-\sum\nolimits_{j=0}^{k-3}\widetilde{g}_{-p+j}^{(k)} & \cdots & 
\widetilde{g}_{k-n+1}^{(k)}-\sum\nolimits_{j=0}^{k-(n-p)}\widetilde{g}_{-p+j}^{(k)} & \widetilde{g}_{k-n}^{(k)}-\sum\nolimits_{j=0}^{k-(n-p+1) }\widetilde{g}_{-p+j}^{(k)}
\\ 
1 & -1 & \cdots & \cdots & -1 \\ 
0 & 1 & \cdots &  & -1 \\ 
\overset{.}{\overset{.}{}} &  &  &  & \vdots \\ 
0 &  & \cdots & 1 & -1
\end{pmatrix}.$}
\end{displaymath}

So we have
\begin{equation*}
F_{(n,k)}^{n-2}=
\begin{pmatrix}
\widetilde{g}_{-n+1}^{(k)} & \widetilde{g}_{-n+k}^{(k)}-\sum\nolimits_{j=0}^{k-3}\widetilde{g}_{-n+2+j}^{(k)} \\ 
1 & -1
\end{pmatrix}.
\end{equation*}

Lemma \ref{Brualdi} gives
\begin{align*}
\per F_{(n,k)}& =\per 
\begin{pmatrix}
\widetilde{g}_{-n+1}^{(k)} & \widetilde{g}_{-n+k}^{(k)}-\sum\nolimits_{j=0}^{k-3}\widetilde{g}_{-n+2+j}^{(k)} \\ 
1 & -1
\end{pmatrix}\\
&=-\widetilde{g}_{-n+1}^{(k)}+\widetilde{g}_{-n+k}^{(k)}-\sum\nolimits_{j=0}^{k-3}\widetilde{g}_{-n+2+j}^{(k)}\\
&=\widetilde{g}_{-n+k}^{(k)}-\left( \widetilde{g}_{-n+1}^{(k)}+\widetilde{g}_{-n+2}^{(k)}+\cdots +\widetilde{g}_{-n+k-1}^{(k)}\right).
\end{align*}
So $\per F_{(n,k)}=\widetilde{g}_{-n}^{(k)}$ and the proof of Identity \eqref{5.2} is complete.

By Lemma \ref{Brualdi}, we can write 
\begin{equation}
\per \widetilde{F}_{(n,k)}=\per \widetilde{F}_{(n,k)}^{1},\label{eq-proof}
\end{equation}
where $\widetilde{F}_{(n,k)}^{1}$ is the first contraction of $\widetilde{F}_{(n,k)}$.

Then $\widetilde{f}_{11}^{1}=\widetilde{f}_{12}^{1}=\cdots =\widetilde{f}
_{1k-1}^{1}=0$, $\widetilde{f}_{1k}^{1}=1$ and $\widetilde{f}_{1q}^{1}=0$ $
(q\geq k+1)$; and for $i=2,\dots ,n-1$, $\widetilde{f}
_{i,i-1}^{1}=1$, $\widetilde{f}_{ij}^{1}=-1$ for $(0\leq j-i\leq k-2)$ and $\widetilde{f}_{ij}^{1}=1$ for $(j-i=k-1)$.

That is, 
$$\per \widetilde{F}_{(n,k)}^{1}=\per
\begin{pmatrix}
0 & 0 & \cdots & 1 & 0 & \cdots & 0 \\ 
1 & -1 & \cdots & -1 & 1 & \ddots & \vdots \\ 
0 & 1 &  &  & \ddots & \ddots & 0 \\ 
&  &  &  &  & \ddots & 1 \\ 
&  &  &  &  &  & -1 \\ 
&  &  &  &  &  & \vdots \\ 
0 &  & \cdots &  & 0 & 1 & -1
\end{pmatrix}
_{n-1}.$$

Computing $\per \widetilde{F}_{(n,k)}^{1}$ by the
Laplace expansion with respect to the first column, we obtain
$$\per \widetilde{F}_{(n,k)}^{1}=\per
\begin{pmatrix}
1 & 0 & \cdots & 0 & 0 & \cdots & 0 \\ 
1 & -1 & \cdots & -1 & 1 & \ddots & \vdots \\ 
&  &  &  & \ddots & \ddots & 0 \\ 
&  &  &  &  & \ddots & 1 \\ 
&  &  &  &  &  & -1 \\ 
&  &  &  & \ddots & \ddots & \vdots \\ 
&  & \cdots &  &  & 1 & -1
\end{pmatrix}
_{n-k}.$$

From identity \eqref{5.2}, $\per \widetilde{F}_{(n,k)}^{1}=\per F_{(n-k,k)}=\widetilde{g}_{-(n-k)}^{(k)}$, and from \eqref{eq-proof}, we have,

$\per \widetilde{F}_{(n,k)}=\widetilde{g}_{-n+k}^{(k)}$. The proof is complete.
\end{proof}

We conclude this section with the proof of Theorem \ref{theorem-second-lucas}.

\begin{proof} If $n=4$,
$$\per B_{4}=\per
\begin{pmatrix}
1 & 1 & 1 & 0 \\ 
1 & 1 & 0 & 1 \\ 
0 & 1 & 1 & 0 \\ 
0 & 0 & -1 & -1
\end{pmatrix}
=-4=l_{-3}^{(2)}.$$ 
By induction on $n$, we assume that $\per B_{n}=l_{-n+1}^{(2)}$ and we establish that $\per B_{n+1}=l_{-n}^{(2)}$.

Let $$F=
\begin{pmatrix}
0 \\ 
\vdots \\ 
0 \\ 
1
\end{pmatrix}.
$$ 
Then $$B_{n+1}=
\begin{pmatrix}
B_{n} & F \\ 
F^{t} & -1
\end{pmatrix}
.$$ 
Let $B_{n}^{(p)}$ be the $p^{\text{th}}$ contraction of $B_{n}$. Since 
$$
B_{n}^{(n-2)}=
\begin{pmatrix}
l_{-n+2}^{(2)} & l_{-n+3}^{(2)} \\ 
1 & -1
\end{pmatrix},
$$ 
we get $$\per B_{n+1}=\per B_{n+1}^{(n-2)}=\per
\begin{pmatrix}
l_{-n+2}^{(2)} & l_{-n+3}^{(2)} & 0 \\ 
1 & -1 & 1 \\ 
0 & 1 & -1
\end{pmatrix}
=l_{-n+2}^{(2)}-l_{-n+3}^{(2)}+l_{-n+2}^{(2)}=l_{-n+2}^{(2)}-l_{-n+1}^{(2)}=l_{-n}^{(2)}.$$
The proof is completed.
\end{proof}

\section{Acknowledgments}
The authors express their gratitude to the reviewer and the editor for the helpful remarks and recommendations which improved the quality of the paper. This work is supported partially by DGRSDT Grant number C0656701.


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\end{thebibliography}

\newpage

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\noindent 2020 {\it Mathematics Subject Classification}:
Primary 15A15; Secondary 11B39, 11B37, 05A05.

\noindent \emph{Keywords: }
permanent, Fibonacci number, Lucas number, Pell number, Jacobsthal number, order-$k$ Fibonacci number, negatively subscripted Fibonacci number, Hessenberg matrix.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences \seqnum{A077990}, \seqnum{A078021}, \seqnum{A078064}, \seqnum{A215936}, and \seqnum{A276229}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received December 20 2021; 
revised versions received December 22 2021; June 16 2022; 
June 19 2022.
Published in {\it Journal of Integer Sequences}, June 22 2022.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{https://cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in


\end{document}

                                                                                



