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\begin{center}
\vskip 1cm{\LARGE\bf On generalized Leonardo $p$-numbers}
\vskip 1cm
\large
Kantaphon Kuhapatanakul and Pongpol Ruankong\\
Department of Mathematics\\
Faculty of Science\\
Kasetsart University\\
Bangkok 10900\\
Thailand\\
\href{mailto:fscikpkk@ku.ac.th}{\tt fscikpkk@ku.ac.th} \\
\href{mailto:fscippru@ku.ac.th}{\tt fscippru@ku.ac.th}\\
\end{center}

\vskip .2 in
\begin{abstract}
We introduce a generalization of Leonardo $p$-numbers, which extends the definition of Leonardo $p$-numbers given by Tan and Leung, and investigate some of their basic properties. Moreover, we give a companion matrix and derive some identities regarding the numbers.
\end{abstract}

\section{Introduction}

The sequence of the Fibonacci numbers \seqnum{A000045} $\{F_n\}_{n\ge0}$ is the second-order linear recurrence sequence given by 
$$F_0=0, F_1=1 \ \ \ \text{and} \ \ \ \ F_{n+1}=F_n+F_{n-1}\ \ \mbox{for}\ n\ge1.$$

One of the generalizations of Fibonacci numbers, given by Stakhov \cite{st}, is the Fibonacci $p$-numbers $F_p(n)$, which is defined for positive integers $p$ by 
$$F_p(n) = F_p(n-1) + F_p(n-p-1)\ \ \mbox{for}\ n&gt;p$$
with the initial values $F_p(0)=0$ and $F_p(1)=F_p(2)=\dots=F_p(p)=1.$

Furthermore, Catarino and Borges in \cite{cb1} introduced the Leonardo sequence $\{\text{Le}_n\}_{n\ge0}$ and derived some of its properties. It is the sequence \seqnum{A001595} satisfying the recurrence relation
$$\text{Le}_0=\text{Le}_1=1 \ \ \ \text{and} \ \ \ \ \text{Le}_{n}=\text{Le}_{n-1}+\text{Le}_{n-2}+1\ \ \mbox{for}\ n\ge 2.$$

The first author and Chobsorn \cite{kc} presented the generalized Leonardo sequence and obtained some properties of this sequence. Recently, Tan and Leung \cite{tl} have introduced the sequence of the Leonardo $p$-numbers $\{{\cal L}_{p,n}\}_{n\ge0}$ as
$${\cal L}_p(n) = {\cal L}_p(n-1) + {\cal L}_p(n-p-1)+p\ \ \mbox{for}\ n&gt;p$$
with the initial values ${\cal L}_p(0)={\cal L}_p(1)=\dots={\cal L}_p(p)=1.$ When $p=1$, the Leonardo $p$-numbers reduce to the Leonardo numbers. 

In this paper, we extend the aforementioned definition of the Leonardo $p$-numbers and derive some properties of the new sequence. 

\section{Generalized Leonardo $p$-numbers}

We begin by defining a sequence of the generalized Leonardo $p$-numbers and establishing its basic properties.
\begin{definition}
For all fixed pairs of positive integers $p$ and $k$, define the generalized Leonardo $p$-sequence $\{{\cal L}_{p,k}(n)\}_{n\ge0}$ by
$${\cal L}_{p,k}(n)={\cal L}_{p,k}(n-1)+{\cal L}_{p,k}(n-p-1)+k\ \ \mbox{for}\ n&gt;p$$
with the initial values ${\cal L}_{p,k}(0)={\cal L}_{p,k}(1)=\dots={\cal L}_{p,k}(p)=1$.
\end{definition}

Some special cases of the generalized Leonardo $p$-numbers are as follows.
\begin{itemize}
   \item  If $k=p$,  the generalized Leonardo $p$-numbers are the Leonardo $p$-numbers.
   \item  If $p=1$,  the generalized Leonardo $p$-numbers are the generalized Leonardo numbers.
   \item  If $p=1$ and $k=2$, this sequence reduces to the sequence \seqnum{A111314}.
\end{itemize}

We compute the first few terms of the sequences $F_2(n)$ and ${\cal L}_{2,k}(n)$ for $k=1,2,3,4,5$.

\begin{table} [H]
\begin{center}
\begin{tabular} {|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$n$  &amp;1  &amp;2  &amp;3  &amp;4  &amp;5  &amp;6  &amp;7  &amp;8  &amp;9  &amp;10 &amp;11 &amp;12  &amp;13  &amp;14  \\
\hline
$F_2(n)$    &amp;1  &amp;1  &amp;1  &amp;2  &amp;3  &amp;4  &amp;6  &amp;9  &amp;13  &amp;19 &amp;28 &amp;41  &amp;60  &amp;88\\
\hline
${\cal L}_{2,1}(n)$ &amp;1  &amp;1  &amp;3  &amp;5  &amp;7  &amp;11  &amp;17  &amp;25  &amp;37 &amp;55 &amp;81  &amp;119  &amp;175  &amp;257\\
\hline
${\cal L}_{2,2}(n)$ &amp;1  &amp;1 &amp;4  &amp;7  &amp;10  &amp;16  &amp;25  &amp;37 &amp;55  &amp;82 &amp;121 &amp;178  &amp;262  &amp;385\\
\hline
${\cal L}_{2,3}(n)$ &amp;1  &amp;1  &amp;5  &amp;9  &amp;13  &amp;21  &amp;33  &amp;49  &amp;73  &amp;109 &amp;161 &amp;237  &amp;349  &amp;513\\
\hline
${\cal L}_{2,4}(n)$ &amp;1  &amp;1  &amp;6  &amp;11  &amp;16  &amp;26  &amp;41  &amp;61  &amp;91  &amp;136  &amp;201 &amp;296  &amp;436  &amp;641\\
\hline
${\cal L}_{2,5}(n)$ &amp;1  &amp;1  &amp;7  &amp;13  &amp;19  &amp;31  &amp;49  &amp;73  &amp;109  &amp;163  &amp;241 &amp;355  &amp;523  &amp;769\\
\hline
\end{tabular} 
\caption{\label{tab1} The first $14$ terms of $F_2(n), {\cal L}_{2,1}(n), {\cal L}_{2,2}(n),{\cal L}_{2,3}(n),{\cal L}_{2,4}(n)$ and ${\cal L}_{2,5}(n)$.}
\end{center}
\end{table}

The Fibonacci $2$-numbers $F_2(n)$ and the generalized Leonardo $2$-numbers ${\cal L}_{2,2}(n)$ are the sequences \seqnum{A000930} and \seqnum{A362255}, respectively.

The non-homogeneous recurrence relation of the generalized Leonardo $p$-numbers can be rewritten as the following homogeneous recurrence relation:
$${\cal L}_{p,k}(n)={\cal L}_{p,k}(n-1)+{\cal L}_{p,k}(n-p)-{\cal L}_{p,k}(n-2p-1)\ \ \mbox{for}\  n&gt;2p.$$


Next, we establish a connection between the generalized Leonardo $p$-numbers and the Fibonacci $p$-numbers.
\begin{theorem} \label{gl1}
For positive integers $p, k$ and a nonnegative integer $n$, we have
\begin{equation}   \label{lf1}
   {\cal L}_{p,k}(n)=(k+1)F_p(n+1)-k. \end{equation}
\end{theorem}
\begin{proof}
It is easy to check that the identity \eqref{lf1} holds for $n=0,1,2,\dots,p$. Assume that, for some $n\ge p$, the identity \eqref{lf1} holds for all integers $m$ such that $0 \le m \le n$. By the definition and the inductive hypothesis, we obtain
\begin{align*}
     {\cal L}_{p,k}(n+1)&amp;={\cal L}_{p,k}(n)+{\cal L}_{p,k}(n-p)+k\\
                                 &amp;=(k+1)F_p(n+1)-k+(k+1)F_p(n-p+1)-k+k\\
                                 &amp;=(k+1)F_p(n+2) -k,
\end{align*}
which shows that the identity \eqref{lf1} holds for $n+1$, thereby proving the theorem.
\end{proof}

\begin{theorem}  \label{sl1}
For positive integers $p, k$ and a nonnegative integer $n$, we have
$$\sum_{i=0}^n{\cal L}_{p,k}(i)={\cal L}_{p,k}(n+p+1)-k(n+1)-1.$$
\end{theorem}
\begin{proof}
By the definition of ${\cal L}_{p,k}(n)$, write
$${\cal L}_{p,k}(i)={\cal L}_{p,k}(i+p+1)-{\cal L}_{p,k}(i+p)-k.$$
Summing a telescoping series, we get
$$\sum_{i=0}^n{\cal L}_{p,k}(i)= {\cal L}_{p,k}(n+p+1)-{\cal L}_{p,k}(p)-k(n+1).$$
Since ${\cal L}_{p,k}(p)=1$, we obtain the desired result.
\end{proof}

\begin{theorem}  \label{sl2}
Let $m$ and $n$ be nonnegative integers. For positive integers $p, k$, we have
$$\sum_{i=0}^n{\cal L}_{p,k}((p+1)i+m)=\begin{cases}
                  {\cal L}_{p,k}\left((p+1)n+m+1\right)-kn,   &amp;\text{if}\ 0\le m&lt;p;\\
                  {\cal L}_{p,k}\left((p+1)(n+1)\right)-k(n+1)-1, &amp;\text{if}\ m=p.
                  \end{cases}$$
\end{theorem}
\begin{proof}
Since
$${\cal L}_{p,k}(i(p+1)+m)={\cal L}_{p,k}\left(i(p+1)+m+1\right)-{\cal L}_{p,k}\left((i-1)(p+1)+m+1\right)-k,$$
by summing a telescoping series we get
$$\sum_{i=0}^n{\cal L}_{p,k}(i(p+1)+m)={\cal L}_{p,k}\left((p+1)n+m+1\right)-{\cal L}_{p,k}(-p+m)-k(n+1).$$
Since ${\cal L}_{p,k}(-i)=-k$ for $1\le i\le p$ and ${\cal L}_{p,k}(0)=1$, the result follows.
\end{proof}

Here are some examples of Theorem \ref{sl2} where $p=2$.
\begin{itemize}   
  \item [$(i)$]  $\displaystyle\sum_{i=0}^n{\cal L}_{2,k}(3i)={\cal L}_{2,k}\left(3n+1\right)-kn$
   \item [$(ii)$]  $\displaystyle\sum_{i=0}^n{\cal L}_{2,k}(3i+1)={\cal L}_{2,k}\left(3n+2\right)-kn$
   \item [$(iii)$]  $\displaystyle\sum_{i=0}^n{\cal L}_{2,k}(3i+2)={\cal L}_{2,k}\left(3n+3\right)-k(n+1)-1.$
\end{itemize}

Moreover, substituting $k=p$ in Theorem \ref{gl1}, Theorem \ref{sl1} and Theorem \ref{sl2}, we obtain the identities of Theorem $1$, Proposition $2$ and Theorem $2$, in \cite{tl}, respectively.

\section{Main results}

In this section, we introduce a companion matrix of the generalized Leonardo $p$-numbers for $p\ge2$ (for the case $p=1$, see \cite[page $4$]{kc}). For a given integer $p\ge2$, let $H(p)$ consist of all those $(2p+1)^\text{th}$ order recurrence sequences $\{{\cal H}_p(n)\}_{n\in\mathbb Z}$ of real numbers satisfying the relation
$${\cal H}_p(n)={\cal H}_p(n-1)+{\cal H}_p(n-p)-{\cal H}_p(n-2p-1).$$
Note that the sequences ${\cal L}_{p,k}(n)$ are elements of $H(p)$.
 
Two significant sequences $A_p$ and $B_p$ in $H(p)$ are defined with the following initial values:
\begin{itemize}
  \item [$(i)$]  $A_p(1)=A_p(2)=\cdots=A_p(p)=1$ and $A_p(0)=A_p(-1)=\cdots=A_p(-p)=0,$
  \item [$(i)$]  $B_p(1)=B_p(2)=\cdots=B_p(p)=1$, $B_p(0)=B_p(-1)=\cdots=B_p(-p+2)=0$, $B_p(-p+1)=1$ and $B_p(-p)=0$.
\end{itemize}
  
We first derive a relationship between  the sequences $F_p$ and $A_p$.
\begin{theorem}
Let $n$ be a positive integer. For positive integers $p$, we have 
\begin{equation}    \label{FA}
   F_p(n)=A_p(n)-A_p(n-p).
\end{equation}
\end{theorem}
\begin{proof}
Observe that $$F_p(n)=\begin{cases}
                                    A_p(n),       \ \ &amp;\text{if}\ 0\le n\le p;\\
                                    A_p(n)-1,  \ &amp;\text{if}\ p&lt;n\le 2p.
                                  \end{cases}$$
So the identity \eqref{FA} holds for all integers $n=0,1,2,\dots,2p$. Now, assume that for some integer $n\ge 2p$, the identity \eqref{FA} holds for all integers $m$ such that $0\le m \le n$. By the definition and the inductive hypothesis, 
\begin{align*}
     A_p(n+1)-A_p(n-p+1) &amp;= A_p(n)+A_p(n-p+1)-A_p(n-2p)-A_p(n-p+1)\\
                                     &amp;= A_p(n)-A_p(n-p)+A_p(n-p)-A_p(n-2p)\\
                                     &amp;= F_p(n)+F_p(n-p)\\
                                     &amp;= F_p(n+1),
\end{align*} 
showing that the identity \eqref{FA} holds for $n+1$.
\end{proof}

Consequently, we can write $A_p(n)$ as a sum of the Fibonacci $p$-numbers.
\begin{corollary}
Let $j$ be an integer such that $0\le j&lt;p$. Then 
\begin{equation}    \label{FA2}
   A_p(mp+j)=\sum_{i=0}^mF_p(ip+j).
\end{equation}
\end{corollary}

Some special cases of the sequences $A_p$ and $B_p$ and their first few terms are given in the tables. 

$\bullet$ For $p=2$, the first 16 terms of $A_2(n)$ and $B_2(n)$ are
\begin{center}
\begin{tabular} {|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$n$ &amp;1  &amp;2  &amp;3  &amp;4  &amp;5  &amp;6  &amp;7  &amp;8  &amp;9  &amp;10 &amp;11 &amp;12  &amp;13  &amp;14  &amp;15  &amp;16\\
\hline
$A_2(n)$ &amp;1  &amp;1  &amp;2  &amp;3  &amp;5  &amp;7  &amp;11  &amp;16  &amp;24  &amp;35 &amp;52 &amp;76  &amp;112  &amp;164  &amp;241  &amp;353\\
\hline
$B_2(n)$ &amp;1  &amp;1  &amp;2  &amp;2  &amp;4  &amp;5  &amp;8  &amp;11  &amp;17  &amp;24 &amp;36 &amp;52  &amp;77  &amp;112  &amp;165  &amp;241\\
\hline
\end{tabular}
\end{center}

$\bullet$ For $p=3$, the first 17 terms of $A_3(n)$ and $B_3(n)$ are
\begin{center}
\begin{tabular} {|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$n$ &amp;1  &amp;2  &amp;3  &amp;4  &amp;5  &amp;6  &amp;7  &amp;8  &amp;9  &amp;10 &amp;11 &amp;12  &amp;13  &amp;14  &amp;15  &amp;16 &amp;17\\
\hline
$A_3(n)$ &amp;1  &amp;1  &amp;1 &amp;2  &amp;3  &amp;4  &amp;6  &amp;8  &amp;11  &amp;16  &amp;22 &amp;30 &amp;42  &amp;58  &amp;80  &amp;111  &amp;153\\
\hline
$B_3(n)$ &amp;1  &amp;1  &amp;1 &amp;2  &amp;2  &amp;3  &amp;5  &amp;6  &amp;8  &amp;12  &amp;16 &amp;22 &amp;31  &amp;42  &amp;58  &amp;81   &amp;111\\
\hline
\end{tabular}
\end{center}

$\bullet$ For $p=4$, the first 18 terms of $A_4(n)$ and $B_4(n)$ are
\begin{center}
\begin{tabular} {|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$n$ &amp;1  &amp;2  &amp;3  &amp;4  &amp;5  &amp;6  &amp;7  &amp;8  &amp;9  &amp;10 &amp;11 &amp;12  &amp;13  &amp;14  &amp;15  &amp;16 &amp;17  &amp;18\\
\hline
$A_4(n)$ &amp;1  &amp;1  &amp;1 &amp;1 &amp;2  &amp;3  &amp;4  &amp;5  &amp;7  &amp;9  &amp;12  &amp;16 &amp;22 &amp;29  &amp;38  &amp;50  &amp;67  &amp;89\\
\hline
$B_4(n)$ &amp;1  &amp;1  &amp;1 &amp;1 &amp;2  &amp;2  &amp;3  &amp;4  &amp;6  &amp;7  &amp;9  &amp;12 &amp;17 &amp;22  &amp;29  &amp;38  &amp;51   &amp;67\\
\hline
\end{tabular}
\end{center}

The sequences $A_2, A_4, B_3$ and $B_4$ are the sequences \seqnum{A023435}, \seqnum{A233522}, \seqnum{A017819} and \seqnum{A017830}, respectively.

Some additional properties of the sequences $A_p$ and $B_p$ are given below. The proof is a straightforward induction and shall be omitted.
\begin{proposition}
Let $n$ be a nonnegative integer. Then
\begin{itemize}
  \item [$(i)$]   $A_p(n)=\begin{cases}
                      A_p(n-1)+A_p(n-p-1)+1,\ \ &amp;\text{if} \ n\equiv 1 \pmod p ;\\ 
                      A_p(n-1)+A_p(n-p-1), \ \ \ \ \ &amp; \text{otherwise}.
                     \end{cases}$
 \item [$(ii)$]   $B_p(n)=\begin{cases}
                      B_p(n-1)+B_p(n-p-1)+1,\ \ &amp;\text{if} \ n\equiv 1 \pmod p ;\\ 
                      B_p(n-1)+B_p(n-p-1)-1,\ \ &amp;\text{if} \ n\equiv 2 \pmod p ;\\ 
                      B_p(n-1)+B_p(n-p-1), \ \ \ \ \ &amp;\text{otherwise}.
                     \end{cases}$\\
\end{itemize}
\end{proposition}

Next, we derive a relationship between the sequences $A_p$ and $B_p$.
\begin{proposition}
Let $n\ge0$ be an integer. Then
\begin{equation}  \label{ab}
     A_p(n)=\begin{cases}
                      B_p(n+1)-1,\ \ \ \text{if} \ n\equiv 0 \pmod p ;\\ 
                      B_p(n+1),\ \ \ \ \ \ \ \ \text{otherwise}. 
                     \end{cases}
\end{equation}
\end{proposition}
\begin{proof}
It is easy to see that the base cases of the identity \eqref{ab} hold. Now assume that, for some integer $n\ge 2p$, the identity \eqref{ab} holds for all integers $m$ such that $0\le m \le n$. By the definition and the inductive hypothesis, we obtain
\begin{align*}
     A_p&amp;(n+1)=A_p(n)+A_p(n-p+1)-A_p(n-2p)\\
                   &amp;=\begin{cases}
                       B_p(n+1)-1+B_p(n-p+2)-B_p(n-2p+1)+1, \ \ &amp;\text{if}\ n\equiv 0\ (\text{mod}\ p) ;\\
                       B_p(n+1)+B_p(n-p+2)-1-B_p(n-2p+1), \ \ &amp;\text{if}\ n \equiv -1\ (\text{mod}\ p) ;\\
                       B_p(n+1)+B_p(n-p+2)-B_p(n-2p), \ \ &amp;\text{otherwise} 
                        \end{cases}\\
                   &amp;=\begin{cases}
                       B_p(n+2)-1, \ \ \ \  &amp;\text{if}\ n+1\equiv 0\ (\text{mod}\ p) ;\\
                       B_p(n+2), \  &amp;\text{otherwise} , 
                        \end{cases}
\end{align*}
thereby proving the theorem.
\end{proof}

Now we proceed to the main result. For $n\ge p\ge2$, define the $(2p+1)\times (2p+1)$ companion matrix $Q_p=[q_{ij}]$ where
$$q_{ij}=\begin{cases}
               1, \ \ \ \ \ \ &amp;\text{if} \ i=1,p \ \text{and} \ j=1,\ \text{and}\ i=j+1 ;\\
               -1,  \ \ \ \  &amp;\text{if} \ i=2p+1, j=1 ;\\
               0,   \ \ \ \ \ \ &amp;\text{otherwise}.
             \end{cases}$$
For instance, the matrices $Q_p$ for $p=2,3,4$ are as follows:
$$Q_2=\begin{bmatrix}
      1    &amp;1    &amp;0     &amp;0    &amp;0\\
      1    &amp;0    &amp;1     &amp;0    &amp;0\\
      0    &amp;0    &amp;0     &amp;1    &amp;0\\
      0    &amp;0    &amp;0     &amp;0    &amp;1\\
      -1   &amp;0    &amp;0    &amp;0     &amp;0
 \end{bmatrix},\ \ 
Q_3=\begin{bmatrix}
      1    &amp;1    &amp;0     &amp;0    &amp;0    &amp;0    &amp;0\\
      0    &amp;0    &amp;1     &amp;0    &amp;0    &amp;0    &amp;0\\
      1    &amp;0    &amp;0     &amp;1    &amp;0    &amp;0    &amp;0\\
      0    &amp;0    &amp;0     &amp;0    &amp;1    &amp;0    &amp;0\\
      0    &amp;0    &amp;0     &amp;0    &amp;0    &amp;1    &amp;0\\
      0    &amp;0    &amp;0     &amp;0    &amp;0    &amp;0    &amp;1\\
      -1   &amp;0    &amp;0    &amp;0     &amp;0    &amp;0    &amp;0\\
 \end{bmatrix},$$
$$Q_4=\begin{bmatrix}
      1    &amp;1    &amp;0     &amp;0    &amp;0    &amp;0    &amp;0    &amp;0    &amp;0\\
      0    &amp;0    &amp;1     &amp;0    &amp;0    &amp;0    &amp;0    &amp;0    &amp;0\\
      0    &amp;0    &amp;0     &amp;1    &amp;0    &amp;0    &amp;0    &amp;0    &amp;0\\
      1    &amp;0    &amp;0     &amp;0    &amp;1    &amp;0    &amp;0    &amp;0    &amp;0\\
      0    &amp;0    &amp;0     &amp;0    &amp;0    &amp;1    &amp;0    &amp;0    &amp;0\\
      0    &amp;0    &amp;0     &amp;0    &amp;0    &amp;0    &amp;1    &amp;0    &amp;0\\
      0    &amp;0    &amp;0     &amp;0    &amp;0    &amp;0    &amp;0    &amp;1    &amp;0\\
      0    &amp;0    &amp;0     &amp;0    &amp;0    &amp;0    &amp;0    &amp;0    &amp;1\\
      -1   &amp;0    &amp;0    &amp;0     &amp;0    &amp;0    &amp;0    &amp;0    &amp;0\\
 \end{bmatrix}.$$

\begin{theorem}  \label{prodQ}
Let $n$ be a nonnegative integer. Then
$$Q_p^n=\begin{bmatrix}
      A_p(n+1)     &amp;A_p(n)            &amp;A_p(n-1)       &amp;\cdots    &amp;A_p(n-2p+1)\\
      B_p(n-p+2)  &amp;B_p(n-p+3)    &amp;B_p(n-p+4)    &amp;\cdots    &amp;B_p(n-3p+2)\\
      B_p(n-p+3)  &amp;B_p(n-p+4)    &amp;B_p(n-p+5)    &amp;\cdots    &amp;B_p(n-3p+3)\\
      \vdots          &amp; \vdots          &amp;  \ddots          &amp;             &amp;\vdots\\
      B_p(n)         &amp;B_p(n-1)        &amp;B_p(n-2)        &amp;\cdots    &amp;B_p(n-2p)\\ 
      -A_p(n-p)     &amp;-A_p(n-p-1)    &amp;A_p(n-p-2)     &amp;\cdots    &amp;A_p(n-3p)\\
      \vdots          &amp; \vdots          &amp;  \ddots          &amp;             &amp;\vdots\\
      -A_p(n-1)     &amp;-A_p(n-2)      &amp;A_p(n-3)         &amp;\cdots    &amp;A_p(n-2p-1)\\
      -A_p(n)        &amp;-A_p(n-1)      &amp;A_p(n-2)         &amp;\cdots    &amp;A_p(n-2p)
 \end{bmatrix}.$$ 
\end{theorem}
\begin{proof}
The case $n=1$ follows from the initial values of the sequences $A_p$ and $B_p$. Write
$$Q_p^{n+1}=Q_p^nQ_p.$$
By the inductive hypothesis and the definitions of $A_p$ and $B_p$, a straightforward matrix multiplication shows that the result holds for $n+1$.
\end{proof}

For positive integers $m$, define the $(2p+1)\times (2p+1)$ matrix $R_{p,m}$ by
$$R_{p,m}=\begin{bmatrix}
  {\cal H}_p(m)     &amp;{\cal H}_p(m-1)   &amp;{\cal H}_p(m-2)   &amp;\cdots   &amp;{\cal H}_p(m-2p)\\
  {\cal H}_p(m-1)  &amp;{\cal H}_p(m-2)   &amp;{\cal H}_p(m-3)   &amp;\cdots   &amp;{\cal H}_p(m-2p-1)\\
  {\cal H}_p(m-2)  &amp;{\cal H}_p(m-3)   &amp;{\cal H}_p(m-4)   &amp;\cdots   &amp;{\cal H}_p(m-2p-2)\\
    \vdots            &amp; \vdots               &amp; \ddots                &amp;          &amp;\vdots\\
  {\cal H}_p(m-2p)  &amp;{\cal H}_p(m-2p-1)  &amp;{\cal H}_p(m-2p-2)  &amp;\cdots   &amp;{\cal H}_p(m-4p)
 \end{bmatrix}$$ where ${\cal H}_p$ is an element of $H(p)$.

\begin{theorem}
Let $n\ge0$ be an integer. For every positive integer $m$, we have
\begin{equation}    \label{rq}
R_{p,m}Q_p^n=R_{p,m+n}.
\end{equation}
\end{theorem}
\begin{proof}
The case $n=0$ is trivial. Now assume that the identity \eqref{rq} is true for an integer $n\ge0$. Since
$$R_{p,m}Q_p^{n+1}=\left(R_{p,m}Q_p^n\right)Q_p,$$
by the inductive hypothesis and the definition of ${\cal H}_p$, a straightforward matrix multiplication shows that the result holds for $n+1$.
\end{proof}

Equating the $(1,1)$-entry on both sides of the equation \eqref{rq}, we obtain the following corollary. 
\begin{corollary} \label{convo}
Let $m,n&gt;0$ be two integers. For a sequence ${\cal H}_p\in H(p)$, we have
$${\cal H}_p(m+n)= {\cal H}_p(m)A_p(n+1)+\sum_{i=1}^{p-1}{\cal H}_p(m-i)B_p(n-p+i+1)-\sum_{i=0}^p{\cal H}_p(m-p-i)A_p(n-p+i).$$
\end{corollary}

Note that, since ${\cal L}_{p,k},A_p,B_p\in H(p)$, the identity of Corollary \ref{convo} holds for each of these sequences as well.

Lastly, we derive the analogous version of Honsberger's formula for the generalized Leonardo $p$-numbers.
\begin{lemma}   \label{sumf}
Let $n$ be a positive integer. Then
$$(k+1)\sum_{i=0}^pF_p(n+i)={\cal L}_{p,k}(n+2p-1)+k.$$
\end{lemma}
\begin{proof}
Using the sum of Fibonacci $p$-numbers formula (see, e.g., \cite{st})
$$\sum_{i=1}^nF_p(i)=F_p(n+p+1)-1,$$ 
we get that
\begin{align*}
\sum_{i=0}^pF_p(n+i)&amp;=\sum_{i=1}^{n+p}F_p(i)-\sum_{i=1}^{n-1}F_p(i)\\
                                 &amp;=F_p(n+2p+1)-F_p(n+p)\\
                                 &amp;=F_p(n+2p).
\end{align*}
Using Theorem \ref{gl1}, we obtain
$$(k+1)\sum_{i=0}^pF_p(n+i)=(k+1)F_p(n+2p)={\cal L}_{p,k}(n+2p-1)+k,$$
as desired.
\end{proof}

\begin{theorem}  \label{hon}
Let $m$ and $n$ be two positive integers. Then 
\begin{align*}  
{\cal L}_{p,k}(m)&amp;{\cal L}_{p,k}(n+1)+\sum_{i=1}^p{\cal L}_{p,k}(m-j){\cal L}_{p,k}(n-p+i)\\
&amp;=(k+1){\cal L}_{p,k}(m+n+1)-k({\cal L}_{p,k}(m+p)+{\cal L}_{p,k}(n+p+1))+pk^2+k.
\end{align*}
\end{theorem}
\begin{proof}
By Theorem \ref{gl1}, write
$$
{\cal L}_{p,k}(m){\cal L}_{p,k}(n)=(k+1)^2F_p(m+1)F_p(n+1)-k(k+1)(F_p(m+1)+F_p(n+1))+k^2.$$
Applying Lemma \ref{sumf} and Honsberger's formula for the Fibonacci $p$-numbers \cite{tk}, we have
\begin{align*}  
{\cal L}_{p,k}(m)&amp;{\cal L}_{p,k}(n+1)+\sum_{i=1}^p{\cal L}_{p,k}(m-j){\cal L}_{p,k}(n-p+i)\\
&amp;=(k+1)^2\left(F_p(m+1)F_p(n+2)+\sum_{i=1}^pF_p(m-i+1)F_p(n-p+i+1)\right)\\
&amp;\ \ -k(k+1)\left(F_p(m+1)+F_p(n+2)+\sum_{i=1}^p F_p(m-i+1)F_p(n-p+i+1)\right)\\
&amp;\ \ +(p+1)k^2\\
&amp;=(k+1)^2F_p(m+n+2)-k(k+1)\left(\sum_{i=0}^pF_p(m-i+1)+\sum_{i=1}^{p+1}F_p(n-p+i+1)\right)\\
&amp;\ \ +(p+1)k^2\\
&amp;=(k+1)({\cal L}_{p,k}(m+n+1)+k)-k\left({\cal L}_{p,k}(m+p)+{\cal L}_{p,k}(n+p+1)+2k\right)\\
&amp;\ \ +(p+1)k^2\\
&amp;=(k+1){\cal L}_{p,k}(m+n+1)-k({\cal L}_{p,k}(m+p)+{\cal L}_{p,k}(n+p+1))+pk^2+k,
\end{align*}
as desired.
\end{proof}

Substuting $k=p$ in Theorem \ref{hon} and using Lemma \ref{sumf}, we obtain Theorem $1$ of \cite{tl}.

\section{Acknowledgment}
The authors are truly grateful to the anonymous referee for his/her helpful comments and suggestions.

\begin{thebibliography}{9} 
\bibitem{ak}
Y. Alp and E. G. Kocer, Some properties of Leonardo numbers, {\em Konuralp J. Math.} {\bf 9} (2021), 183--189.
\bibitem{cb1}
P. Catarino and A. Borges, On Leonardo numbers, {\em Acta Math. Univ. Comenian. (N.S.)} {\bf 89} (2019), 75--86.
\bibitem{cb2}
P. Catarino and A. Borges, A note on incomplete Leonardo numbers, {\em Integers} {\bf20} (2020), \#A43.
\bibitem{kc}
K. Kuhapatanakul and J. Chobsorn, On the generalized Leonardo numbers, {\em Integers} {\bf22} (2022), \#A48.
\bibitem{sh}
A. G. Shannon, A note on generalized Leonardo numbers, {\em Notes Number Theory Discrete Math.} {\bf 25} (2019), 97--101. 
\bibitem{sa}
N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, 2024. Available at \url{https://oeis.org}.
\bibitem{st}
A. P. Stakhov, Fibonacci matrices, a generalization of the Cassini formula, and a new coding theory, {\em Chaos Solitons Fractals} {\bf 30} (2006), 56--66.
\bibitem{tl}
N. Tuglu, E. G. Kocer, and A. Stakhov, Bivariate Fibonacci like $p$-polynomials, {\em Appl. Math. Comput.} {\bf 217} (2011),  10239--10249.
\bibitem{tk}
E. Tan and H. H. Leung, On Leonardo $p$-numbers, {\em Integers} {\bf23} (2023), \#A7.
\end{thebibliography}

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\noindent 2020 {\it Mathematics Subject Classification}: Primary 11B37; Secondry 11B39.

\noindent \emph{Keywords: } Leonardo number, Fibonacci number,  Leonardo $p$-number.

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\noindent (Concerned with sequences
\seqnum{A000045},
\seqnum{A000930},
\seqnum{A001595},
\seqnum{A017819},
\seqnum{A017830},
\seqnum{A023435},
\seqnum{A111314},
\seqnum{A233522}, and
\seqnum{A362255}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received October 2 2023; 
revised versions received  October 9 2023; March 18 2024; March 21 2024.
Published in {\it Journal of Integer Sequences}, April 11 2024.

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