<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[12pt,reqno]{article}

\usepackage[usenames]{color}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amscd}

\usepackage[colorlinks=true,
linkcolor=webgreen,
filecolor=webbrown,
citecolor=webgreen]{hyperref}

\definecolor{webgreen}{rgb}{0,.5,0}
\definecolor{webbrown}{rgb}{.6,0,0}

\usepackage{color}
\usepackage{fullpage}
\usepackage{float}

\usepackage{psfig}
\usepackage{graphics,amsmath,amssymb}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{latexsym}
\usepackage{epsf}

\usepackage{caption}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\evensidemargin}{.1in}
\setlength{\topmargin}{-.1in}
\setlength{\textheight}{8.4in}

\newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}}
\newcommand{\Y}{\mathscr{Y}}

\begin{document}

\begin{center}
\epsfxsize=4in
\leavevmode\epsffile{logo129.eps}
\end{center}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}


\begin{center}
\vskip 1cm{\LARGE\bf Fibonacci and Lucas Sedenions}
\vskip 1cm
\large
G\"{o}ksal Bilgici\\
Department of Computer Education and Instructional Technology\\
Education Faculty\\
Kastamonu University \\
Kastamonu, 37100\\
Turkey \\
\href{mailto:gbilgici@kastamonu.edu.tr}{\tt gbilgici@kastamonu.edu.tr} \\
\ \\
\"{U}mit Toke\c{s}er and Zafer \"{U}nal\\
Department of Mathematics\\
Faculty of Arts and Sciences \\
Kastamonu University\\
Kastamonu, 37150\\
Turkey \\
\href{mailto:utokeser@kastamonu.edu.tr}{\tt utokeser@kastamonu.edu.tr}\\
\href{mailto:zunal@kastamonu.edu.tr}{\tt zunal@kastamonu.edu.tr}\\
\end{center}

\vskip .2 in

\begin{abstract}
The sedenions form a 16-dimensional non-associative and non-commutative algebra over the set of real numbers. In this paper, we introduce the Fibonacci and Lucas sedenions. We present generating functions and Binet formulas for the Fibonacci and Lucas sedenions, and derive adaptations for some well-known identities of Fibonacci and Lucas numbers.
\end{abstract}


\section{Introduction}
\label{sec:introduction}

Sedenions appear in many areas of science, such as electromagnetic theory and linear gravity. Sedenion algebra, which is usually denoted by $\mathbb{S}$, is a 16-dimensional Cayley--Dickson algebra. Because of their zero divisors, sedenions do not form a composition algebra or a division algebra. They are hyper-complex numbers, similar to quaternions and octonions. Sedenion algebra is a non-associative, non-commutative, and non-alternative but power-associative Cayley--Dickson algebra over the reals. Throughout this paper, we take the basis elements of $\mathbb{S}$ as $\{e_0, e_1, \ldots, e_{15}\}$, where $e_0$ is the unit element and $e_1, \ldots, e_{15}$ are imaginaries. A sedenion $S$ can be written as
$$ S=\sum_{i=0}^{15}a_ie_i $$
where $a_0, a_1, \cdots, a_{15}$ are reals.

Imaeda and Imaeda \cite{ima} defined a sedenion by
$$S=(O_1;O_2) \in \mathbb{S}, ~~~~O_1, O_2 \in \mathbb{O}$$
where $\mathbb{O}$ is the octonion algebra over the reals. As a sedenion is an ordered pair of two octonions, the conjugate of a sedenion $S=(O_1;O_2)$ is defined by $\overline{S}=(O_1;-O_2)$. Under the Cayley--Dickson process, the product of two sedenions $S_1=(O_1;O_2)$ and $S_2=(O_3;O_4)$ is
$$S_1S_2=(O_1O_3+\rho\overline{O_4}O_2;O_2\overline{O_3}+O_4O_1).$$
After choosing the field parameter $\rho=-1$ and the generator $e_8$, Imaeda and Imaeda examined the sedenions. By setting $i \equiv e_i$, where $i=0,1,\cdots, 15$, Cawagas \cite{caw} constructed the following multiplication table for the basis of $\mathbb{S}$.

\begin{center}
{\scriptsize
\begin{tabular} {r|rrrrrrrrrrrrrrrr}
$\cdot$ &amp; 0 &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 \\ \hline
0 &amp; 0 &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 \\
1 &amp; 1 &amp; $-0$ &amp; 3 &amp; $-2$ &amp; 5 &amp; $-4$ &amp; $-7$ &amp; 6 &amp; 9 &amp; $-8$ &amp; $-11$ &amp; 10 &amp; $-13$ &amp; 12 &amp; 15 &amp; $-14$ \\
2&amp;2&amp;$-3$&amp;$-0$&amp;1&amp;6&amp;7&amp;$-4$&amp;$-5$&amp;10&amp;11&amp;$-8$&amp;$-9$&amp;$-14$&amp;$-15$&amp;12&amp;13\\
3&amp;3&amp;2&amp;$-1$&amp;$-0$&amp;7&amp;$-6$&amp;5&amp;$-4$&amp;11&amp;$-10$&amp;9&amp;$-8$&amp;$-15$&amp;14&amp;$-13$&amp;12 \\
4&amp;4&amp;$-5$&amp;$-6$&amp;$-7$&amp;$-0$&amp;1&amp;2&amp;3&amp;12&amp;13&amp;14&amp;15&amp;$-8$&amp;$-9$&amp;$-10$&amp;$-11$\\
5&amp;5&amp;4&amp;$-7$&amp;6&amp;$-1$&amp;$-0$&amp;$-3$&amp;2&amp;13&amp;$-12$&amp;15&amp;$-14$&amp;9&amp;$-8$&amp;11&amp;$-10$\\
6&amp;6&amp;7&amp;4&amp;$-5$&amp;$-2$&amp;3&amp;$-0$&amp;$-1$&amp;14&amp;$-15$&amp;$-12$&amp;13&amp;10&amp;$-11$&amp;$-8$&amp;9\\
7&amp;7&amp;$-6$&amp;5&amp;4&amp;$-3$&amp;$-2$&amp;1&amp;$-0$&amp;15&amp;14&amp;$-13$&amp;$-12$&amp;11&amp;10&amp;$-9$&amp;$-8$\\
8&amp;8&amp;$-9$&amp;$-10$&amp;$-11$&amp;$-12$&amp;$-13$&amp;$-14$&amp;$-15$&amp;$-0$&amp;1&amp;2&amp;3&amp;4&amp;5&amp;6&amp;7\\
9&amp;9&amp;8&amp;$-11$&amp;10&amp;$-13$&amp;12&amp;15&amp;$-14$&amp;$-1$&amp;$-0$&amp;$-3$&amp;2&amp;$-5$&amp;4&amp;7&amp;$-6$\\
10&amp;10&amp;11&amp;8&amp;$-9$&amp;$-14$&amp;$-15$&amp;12&amp;13&amp;$-2$&amp;3&amp;$-0$&amp;$-1$&amp;$-6$&amp;$-7$&amp;4&amp;5\\
11&amp;11&amp;$-10$&amp;9&amp;8&amp;$-15$&amp;14&amp;$-13$&amp;12&amp;$-3$&amp;$-2$&amp;1&amp;$-0$&amp;$-7$&amp;6&amp;$-5$&amp;4\\
12&amp;12&amp;13&amp;14&amp;15&amp;8&amp;$-9$&amp;$-10$&amp;$-11$&amp;$-4$&amp;5&amp;6&amp;7&amp;$-0$&amp;$-1$&amp;$-2$&amp;$-3$\\
13&amp;13&amp;$-12$&amp;15&amp;$-14$&amp;9&amp;8&amp;11&amp;$-10$&amp;$-5$&amp;$-4$&amp;7&amp;$-6$&amp;1&amp;$-0$&amp;3&amp;$-2$\\
14&amp;14&amp;$-15$&amp;$-12$&amp;13&amp;10&amp;$-11$&amp;8&amp;9&amp;$-6$&amp;$-7$&amp;$-4$&amp;5&amp;2&amp;$-3$&amp;$-0$&amp;1\\
15&amp;15&amp;14&amp;$-13$&amp;$-12$&amp;11&amp;10&amp;$-9$&amp;8&amp;$-7$&amp;6&amp;$-5$&amp;$-4$&amp;3&amp;2&amp;$-1$&amp;$-0$\\
\end{tabular} }
\captionof{table} {Multiplication table for the basis of $\mathbb{S}$}
\end{center}

Cariow and Cariowa \cite{car} derived an algorithm for the fast multiplication of two sedenions.

The well-known integer sequences of Fibonacci and Lucas numbers are constructed with the same recurrence relation but different initial conditions. Namely, for $n \geq 2$, Fibonacci numbers satisfy the recurrence relation 
$$F_n=F_{n-1}+F_{n-2}, ~~~~F_0=0,~~ F_1=1$$ 
whereas Lucas numbers satisfy the recurrence relation
$$L_n=L_{n-1}+L_{n-2},~~~~L_0=2,~~ L_1=1.$$ 


The generating functions for the Fibonacci sequence $\bigl(F_n \bigr)_{n\geq0}$ and Lucas sequence $\bigl(L_n \bigr)_{n\geq0}$ are
$$\sum_{n=0}^\infty F_nx^n=\frac{x}{1-x-x^2} ~~\textnormal{and}~~\sum_{n=0}^\infty L_nx^n=\frac{2-x}{1-x-x^2}.$$

The Binet formulas for the Fibonacci and Lucas numbers are
\begin{equation} \label{binetfl}
F_{n}=\frac{\alpha ^{n}-\beta ^{n}}{\alpha -\beta } ~~\textnormal{and}~~ L_{n}=\alpha ^{n}+\beta ^{n}
\end{equation}
where $\alpha =\frac{1+\sqrt{5}}{2}$ and $\beta =\frac{1-\sqrt{5}}{2}$ are
the roots of the characteristic equation of $x^{2}-x-1=0$. The positive
root $\alpha $ is known as the golden ratio (see \cite{kos} for details).

Horadam \cite{hor63} defined Fibonacci and Lucas quaternions as 
$$Q_n=F_n+F_{n+1}e_1+F_{n+2}e_2+F_{n+3}e_3$$
and
$$K_n=L_n+L_{n+1}e_1+L_{n+2}e_2+L_{n+3}e_3$$
respectively, where $F_n$ is the $n$th classical Fibonacci number and $L_n$ is the $n$th classical Lucas number. He also defined generalized Fibonacci quaternions by
$$P_n=H_n+H_{n+1}e_1+H_{n+2}e_2+H_{n+3}e_3$$
where $H_n$ is the $n$th generalized Fibonacci number defined by the recursive relation $H_1=p$, $H_2=p+q$, $H_n=H_{n-1}+H_{n-2}$ ($p$ and $q$ are arbitrary integers). Iyer \cite{iye} described various properties of the Fibonacci quaternions and generalized Fibonacci quaternions. Swammy \cite{swa} also studied these two types of quaternions, and obtained some relations between Fibonacci and Lucas numbers. Harman \cite{har} defined complex Fibonacci numbers by the following two-dimensional recurrence relation
\begin{eqnarray*}
G(n+2,m)&amp;=&amp;G(n+1,m)+G(n,m), \\
G(n,m+2)&amp;=&amp;G(n,m+1)+G(n,m)
\end{eqnarray*}
where $G(0,0)=0$, $G(1,0)=1$, $G(0,1)=i$, and $G(1,1)=1+i$. By extending this idea, Horadam \cite{hor93} defined the recurrence relations
\begin{eqnarray*}
G(h+2,l,m,n)&amp;=&amp;G(h+1,l,m,n)+G(h,l,m,n), \\
G(h,l+2,m,n)&amp;=&amp;G(h,l+1,m,n)+G(h,l,m,n), \\
G(h,l,m+2,n)&amp;=&amp;G(h,l,m+1,n)+G(h,l,m,n), \\
G(h,l,m,n+2)&amp;=&amp;G(h,l,m,n+1)+G(h,l,m,n)
\end{eqnarray*}
with initial conditions
\begin{eqnarray*}
G(0,0,0,0)&amp;=&amp;0, G(1,0,0,0)=1, G(0,1,0,0)=e_1, G(0,0,1,0)=e_2, G(0,0,0,1)=e_3,\\
G(1,1,0,0)&amp;=&amp;1+e_1, \ldots , G(0,0,1,1)=e_2+e_3, \\
G(1,1,1,0)&amp;=&amp;1+e_1+e_2, \ldots , G(0,1,1,1)=e_1+e_2+e_3, \\
G(1,1,1,1)&amp;=&amp; 1+e_1+e_2+e_3
\end{eqnarray*}
and examined quaternion recurrence relations. Halici \cite{hal12} investigated some properties of the Fibonacci and Lucas quaternions, obtaining generating functions, Binet formulas, and certain identities.


Kecilioglu and Akkus \cite{k-a} introduced the Fibonacci and Lucas octonions. These hyper-complex numbers are defined as
$$Q_n=\sum_{s=0}^7 F_{n+s}e_s$$
and
$$T_n=\sum_{s=0}^7 L_{n+s}e_s$$
respectively, where $F_n$ and $L_n$ are the $n$th classical Fibonacci and Lucas numbers. In another study \cite{a_k}, they defined split Fibonacci and Lucas octonions in a similar manner. 

In this paper, following Horadam, Kecilioglu, and Akkus, we define the Fibonacci and Lucas sedenions over the sedenion algebra $\mathbb{S}$. The $n$th Fibonacci sedenion is
\begin{equation}
\widehat{F}_n=\sum_{s=0}^{15} F_{n+s}e_s
\end{equation}
and the $n$th Lucas sedenion is
\begin{equation}
\widehat{L}_n=\sum_{s=0}^{15} L_{n+s}e_s.
\end{equation}

Using the identities $F_{-n}=(-1)^{n+1}F_n$ and $L_{-n}=(-1)^{n}L_n$, we immediately have that
the generalized Fibonacci and Lucas sedenions with negative indexes are
$$\widehat{F}_{-n}=\sum_{s=0}^{15}(-1)^{n+s+1}F_{n-s}e_s$$
and
$$\widehat{L}_{-n}=\sum_{s=0}^{15}(-1)^{n+s}L_{n-s}e_s.$$
 For all integers $n$, we can easily see that
\begin{equation}\label{tan}
\widehat{F}_n=\widehat{F}_{n-1}+\widehat{F}_{n-2} ~~\textnormal{and}~~ \widehat{L}_n=\widehat{L}_{n-1}+\widehat{L}_{n-2}.
\end{equation}
Additionally, for any integer $n$, we have
$$\widehat{L}_n=\widehat{F}_{n-1}+\widehat{F}_{n+1}.$$

\section{Generating functions and Binet formulas for the Fibonacci and Lucas sedenions}

Generating functions for the Fibonacci and Lucas sedenions are given in the next theorem.
\begin{theorem}
The generating functions for the Fibonacci and Lucas sedenions are 
\begin{equation}
\sum_{i=0}^\infty \widehat{F}_ix^i=\frac{\widehat{F}_0+\widehat{F}_{-1}x}{1-x-x^2}\label{gen4}
\end{equation}
and
\begin{equation}
\sum_{i=0}^\infty \widehat{L}_ix^i=\frac{\widehat{L}_0+\widehat{L}_{-1}x}{1-x-x^2} \label{gen5}
\end{equation}
respectively.
\end{theorem}
\begin{proof}
Define $f(x)=\sum_{i=0}^\infty \widehat{F}_ix^i$. Expanding the first two terms of $f(x)$, we have
\begin{equation}\label{gen1}
f(x)=\widehat{F}_0+\widehat{F}_1x+\sum_{i=2}^\infty \widehat{F}_ix^i.
\end{equation}
Multiplying both sides of this equation by $-x$ and $-x^2$, we obtain
\begin{equation}\label{gen2}
-xf(x)=-\widehat{F}_0x-\sum_{i=2}^\infty \widehat{F}_{i-1}x^i
\end{equation}
and
\begin{equation}\label{gen3}
-x^2f(x)=-\sum_{i=2}^\infty \widehat{F}_{i-2}x^i.
\end{equation}
Adding Eqs. (\ref{gen1}), (\ref{gen2}) and (\ref{gen3}) with consideration of the first equation in Eq. (\ref{tan}) gives
$$f(x)=\frac{\widehat{F}_0+(\widehat{F}_1-\widehat{F}_0)x}{1-x-x^2}.$$
Using the identity $\widehat{F}_{-1}=\widehat{F}_1-\widehat{F}_0$, we have Eq. (\ref{gen4}). Similarly, we can obtain Eq. (\ref{gen5}).
\end{proof}

The next theorem gives the Binet formulas for the Fibonacci and Lucas sedenions.
\begin{theorem}
For any integer $n$, the $n$th Fibonacci sedenion is
\begin{equation}\label{1}
\widehat{F}_{n}=\dfrac{\widehat{\alpha}\alpha^{n}-\widehat{\beta}\beta^{n}}{\alpha-\beta}
\end{equation}
and the $n$th Lucas sedenion is
\begin{equation}
\widehat{L}_{n}=\widehat{\alpha}\alpha^{n}+\widehat{\beta}\beta^{n} \label{2}
\end{equation}
where $\widehat{\alpha}=\sum_{s=0}^{15} \alpha^s e_s$ and $\widehat{\beta}=\sum_{s=0}^{15} \beta^s e_s$.
\end{theorem}

\begin{proof}
For the first equation, we have
$$\alpha \widehat{F}_{n}+\widehat{F}_{n-1}=
\sum_{s=0}^{15}\left(  \alpha F_{n+s}+F_{n+s-1}\right)  e_{s}.$$
From the identity $\alpha F_{n}+F_{n-1}=\alpha^{n}$, we obtain
\begin{equation}
\alpha \widehat{F}_{n}+\widehat{F}_{n-1}=\widehat{\alpha} \alpha^{n}. \label{3}
\end{equation}
Similarly, we have
\begin{equation}
\beta \widehat{F}_{n}+\widehat{F}_{n-1}=\widehat{\beta}\beta^{n}. \label{4}
\end{equation}
Subtracting Eq. (\ref{4}) from Eq. (\ref{3}) gives
$$(\alpha-\beta)\widehat{F}_n=\widehat{\alpha}\alpha^{n}-\widehat{\beta}\beta^{n}$$
from which we obtain Eq. (\ref{1}).
Summing Eqs. (\ref{3}) and (\ref{4}), we have
$$ \widehat{F}_{n}+2\widehat{F}_{n-1}=\widehat{\alpha}\alpha^{n}+\widehat{\beta}\beta^{n}$$
and, with the help of the identity $L_n=F_n+2F_{n-1}$, we obtain Eq. (\ref{2}).
\end{proof}

When using the Binet formulas to obtain identities for the Fibonacci and Lucas sedenions, we require $\widehat{\alpha}\widehat{\beta}$, $\widehat{\beta}\widehat{\alpha}$, $\widehat{\alpha}^2$, and $\widehat{\beta}^2$. These products are given in the next lemma.
\begin{lemma} \label{yildiz}
We have
\begin{eqnarray}
\widehat{\alpha}\widehat{\beta}&amp;=&amp; \widehat{L}_{0}-\sqrt{5}(\widehat{F}_0-\omega)\label{7a},\\
\widehat{\beta} \widehat{\alpha}&amp;=&amp; \widehat{L}_{0}+\sqrt{5}(\widehat{F}_0-\omega) \label{8a}, \\
\widehat{\alpha}^2&amp;=&amp;-1505175+\widehat{L}_{0}+\sqrt{5} (-673134+\widehat{F}_0),  \label{5}
\end{eqnarray}
and
\begin{equation}
\widehat{\beta}^2=-1505175+\widehat{L}_{0}-\sqrt{5} (-673134+\widehat{F}_0)  \label{6} 
\end{equation}
where
$$\omega=94e_9+94e_{10}+188e_{11}+282e_{12}-188e_{13}+94e_{14}+893e_{15}.$$
\end{lemma}
\begin{proof}
From the definitions of $\widehat{\alpha}$ and $\widehat{\beta}$, and using Table 1,  we have
\begin{eqnarray*}
\widehat{\alpha}\widehat{\beta}&amp;=&amp;\left(\sum_{n=0}^{15} \alpha^n e_n \right)\left(\sum_{n=0}^{15} \beta^n e_n \right)\\
&amp;=&amp;2+e_1+3e_2+\cdots +1364e_{15}-\sqrt{5}\Big(e_1+e_2+2e_3+\cdots +21e_8\\
&amp;&amp;\hspace{0.8cm}-60e_9-39e_{10}-99e_{11}-138e_{12}+421e_{13}+283e_{14}-283e_{15}\Big)
\end{eqnarray*}
where the final equation gives Eq. (\ref{7a}). The others can be computed similarly.
\end{proof}

This lemma gives us the following useful identity:
\begin{equation}
\widehat{\alpha}\widehat{\beta}+\widehat{\beta}\widehat{\alpha}=2\widehat{L}_0.
\end{equation}

\section{Some identities for the Fibonacci and Lucas sedenions}

There are three well-known identities for Fibonacci and Lucas numbers, namely, Catalan's, Cassini's, and d'Ocagne's identities. The proofs of these identities are based on Binet formulas. We can obtain these types of identities for Fibonacci and Lucas sedenions using the Binet formulas derived above. Catalan's identities for Fibonacci and Lucas sedenions are given in the next theorem.

\begin{theorem}
For any integers $m$ and $n$, we have
\begin{equation}
\widehat{F}_{m+n}\widehat{F}_{m-n}-\widehat{F}_{m}^{2}  =\bigl(  -1\bigr)  ^{m}F_{-n}\bigl(  F_{n}\widehat{L}_0 -L_{n}( \widehat{F}_0-\omega)\bigr) \label{9_1}
\end{equation}
and
\begin{equation}
\widehat{L}_{m+n}\widehat{L}_{m-n}-\widehat{L}_{m}^{2}  =-5\bigl(  -1\bigr)  ^{m}F_{-n}\bigl(  F_{n}\widehat{L}_0 - L_{n}( \widehat{F}_0-\omega)\bigr). \label{9_2}
\end{equation}
\end{theorem}

\begin{proof}
From the Binet formula for Fibonacci sedenions, we have
\begin{eqnarray*}
\widehat{F}_{m+n}\widehat{F}_{m-n}-\widehat{F}_{m}^{2} &amp; =&amp;\dfrac{1}{5}\Bigl(  \bigl(  \widehat{\alpha}
\alpha^{m+n}-\widehat{\beta}\beta^{m+n}\bigr)  \bigl(  \widehat{\alpha}\alpha
^{m-n}-\widehat{\beta}\beta^{m-n}\bigr)  \\
&amp;&amp;\hspace{4cm} -\bigl(  \widehat{\alpha}\alpha^{m}-\widehat{\beta}\beta^{m}\bigr)  ^{2}\Bigr) \\
&amp; =&amp;\frac{1}{5}\Bigl(  (-1)^{m-n+1}\bigl(  \widehat{\alpha} \widehat{\beta}\alpha^{2n}+\widehat{\beta}\widehat{\alpha}\beta^{2n}\bigr)   \\
&amp;&amp; \hspace{4cm} +\bigl(-1\bigr)  ^{m}2\widehat{L}_0  \Bigr).
\end{eqnarray*}
We require Eqs. (\ref{7a}) and (\ref{8a}). Using this equation, we obtain
\begin{eqnarray*}
\widehat{F}_{m+n}\widehat{F}_{m-n}-\widehat{F}_{m}^{2} &amp;=&amp;\frac{1}{5}\Bigl(  (-1)^{m-n+1} \bigl( \bigl( \widehat{L}_0-\sqrt{5}(\widehat{F}_0-\omega ) \bigr)\alpha^{2n} \\
&amp;&amp;\hspace{0.3cm}+\bigl( \widehat{L}_0+\sqrt{5}(\widehat{F}_0-\omega ) \bigr)\beta^{2n} \bigr) +2(-1)^m\widehat{L}_0 \Bigr) \\
&amp;=&amp;\frac{1}{5}\bigl(  (-1)^{m-n+1} \bigl( \widehat{L}_0 L_{2n}-5 (\widehat{F}_0-\omega)F_{2n} \bigr) \\
&amp;&amp;\hspace{5cm}+2(-1)^m\widehat{L}_0 \bigr) \\
&amp;=&amp;\frac{1}{5} (-1)^{m-n+1}\widehat{L}_0 \bigl(L_{2n}-2(-1)^n \bigr)\\
&amp;&amp;\hspace{3cm}-(-1)^{m-n+1}F_{2n}(\widehat{F}_0-\omega). \\
\end{eqnarray*}
Using the identity $5F_n^2=L_{2n}-2(-1)^n$  gives
$$\widehat{F}_{m+n}\widehat{F}_{m-n}-\widehat{F}_{m}^{2}=(-1)^{m+n+1}\bigl( F_n^2\widehat{L}_0-F_{2n}(\widehat{F}_0-\omega) \bigr).$$
With the help of the identity $F_{2n}=F_nL_n$, we have Eq. (\ref{9_1}). Similarly, we can obtain Eq. (\ref{9_2}).
\end{proof}

Taking $n=1$ in this theorem and using the identity $L_n-F_n=2F_{n-1}$, we obtain Cassini's identities for Fibonacci and Lucas
sedenions.

\begin{corollary}
For any integer $m$, we have
\begin{equation}
\widehat{F}_{m+1}\widehat{F}_{m-1}-\widehat{F}_{m}^{2}  =\bigl(  -1\bigr)  ^{m}\bigl(  2  \widehat{F}_{-1}+\omega\bigr) 
\end{equation}
and
\begin{equation}
\widehat{L}_{m+1}\widehat{L}_{m-1}-\widehat{L}_{m}^{2}  =-5\bigl(  -1\bigr)  ^{m}\bigl( 2  \widehat{F}_{-1}+\omega\bigr). 
\end{equation}
\end{corollary}

The following theorem gives d'Ocagne's identities for Fibonacci and Lucas sedenions.
\begin{theorem}
For any integers $n$ and $m$, we have
$$\widehat{F}_{m}\widehat{F}_{n+1}-\widehat{F}_{m+1}\widehat{F}_{n}  =( -1)  ^{n}\bigl(  F_{m-n}\widehat{L}_0+ L_{m-n}(\widehat{F}_0-\omega) \bigr) $$
and
$$\widehat{L}_{m}\widehat{L}_{n+1}-\widehat{L}_{m+1}\widehat{L}_{n}  =-5( -1)  ^{n}\bigl(  F_{m-n}\widehat{L}_0+ L_{m-n}(\widehat{F}_0-\omega) \bigr). $$
\end{theorem}

\begin{proof}
Using the Binet formula for the Fibonacci sedenions gives
\begin{eqnarray*}
\widehat{F}_{m}\widehat{F}_{n+1}-\widehat{F}_{m+1}\widehat{F}_{n} &amp; =&amp;\frac{1}{5}\bigl(  \widehat{\alpha}\alpha^{m}
-\widehat{\beta}\beta^{m}\bigr)  \bigl(  \widehat{\alpha}\alpha^{n+1}-\widehat{\beta}\beta^{n+1}\bigr)  \\
&amp;&amp;\hspace{1cm} -\frac{1}{5}\bigl(  \widehat{\alpha}\alpha^{m+1}-\widehat{\beta}\beta
^{m+1}\bigr)  \bigl(  \widehat{\alpha}\alpha^{n}-\widehat{\beta}\beta^{n}\bigr)\\
&amp; =&amp;\frac{\sqrt{5}}{5}(-1)  ^{n}\bigl(  \widehat{\alpha}\widehat{\beta}\alpha^{m-n}
-\widehat{\beta}\widehat{\alpha}\beta^{m-n}\bigr) .
\end{eqnarray*}
We require Eqs. (\ref{7a}) and (\ref{8a}). Substituting these into the previous equation, we have
\begin{eqnarray*}
\widehat{F}_{m}\widehat{F}_{n+1}-\widehat{F}_{m+1}\widehat{F}_{n} &amp; =&amp;\frac{\sqrt{5}}{5}(-1)^n \Bigl( \bigl( \widehat{L}_0-\sqrt{5}(\widehat{F}_0-\omega)\bigr) \alpha^{m-n} \\
&amp;&amp; \hspace{1.5cm} -\bigl( \widehat{L}_0+\sqrt{5}(\widehat{F}_0-\omega)\bigr) \beta^{m-n} \Bigr) \\
&amp;=&amp;(-1)^n \bigl( F_{m-n}\widehat{L}_0-L_{m-n}(\widehat{F}_0-\omega)\bigr).
\end{eqnarray*}
The second identity in the theorem, i.e., d'Ocagne's identity for the Lucas sedenions, can be proved similarly.
\end{proof}

After deriving these three famous identities, we present some other identities for the Fibonacci and Lucas sedenions.
\begin{theorem}
For any integer $n$, we have
\begin{equation}\label{13}
\widehat{L}_{n}^{2}-\widehat{F}_{n}^{2}=\frac{4}{5} (-1505175+\widehat{L}_0)L_{2n}+4(-673134+\widehat{F}_0)F_{2n}+\frac{12}{5}(-1)^n\widehat{L}_0  
\end{equation}
and
\begin{equation} \label{12}
\widehat{L}_{n}^{2}+\widehat{F}_{n}^{2}=\frac{6}{5} (-1505175+\widehat{L}_0)L_{2n}+6(-673134+\widehat{F}_0)F_{2n}+\frac{8}{5}(-1)^n\widehat{L}_0 .
\end{equation}
\end{theorem}

\begin{proof}
Using the Binet formulas for the Fibonacci and Lucas sedenions, we obtain
\begin{eqnarray*}
\widehat{L}_{n}^{2}-\widehat{F}_{n}^{2} &amp;=&amp;\bigl(  \widehat{\alpha}\alpha^{n}+\widehat{\beta}\beta
^{n}\bigr)  ^{2}-\frac{1}{5}\bigl(  \widehat{\alpha}\alpha^{n}-\widehat{\beta}
\beta^{n}\bigr)  ^{2}\\
&amp;=&amp;\bigl(  \widehat{\alpha}^2\alpha^{2n}+\widehat{\beta}^2\beta^{2n}+\widehat{\alpha}
\widehat{\beta}\alpha^{n}\beta^{n}+\widehat{\beta}\widehat{\alpha}\alpha^{n}\beta^{n}\bigr)  \\
&amp;&amp; \hspace{1cm}-\frac{1}{5}\bigl(  \widehat{\alpha}^2\alpha^{2n}+\widehat{\beta}^2\beta^{2n}
-\widehat{\alpha}\widehat{\beta}\alpha^{n}\beta^{n}-\widehat{\beta}\widehat{\alpha}
\alpha^{n}\beta^{n}\bigr).
\end{eqnarray*}
Substituting Eqs. (\ref{7a}) and (\ref{8a}) into the last equation, we have
\begin{equation}\label{d1}
\widehat{L}_{n}^{2}-\widehat{F}_{n}^{2} =\frac{4}{5}\bigl(  \widehat{\alpha}^2\alpha^{2n}+\widehat{\beta}^2\beta
^{2n}\bigr)  +\frac{12}{5}(  -1)  ^{n}\widehat{L}_0.
\end{equation}
Using Eqs. (\ref{5}) and (\ref{6}), we obtain 
\begin{equation}\label{d2}
\bigl(  \widehat{\alpha}^2\alpha^{2n}+\widehat{\beta}^2\beta^{2n}\bigr)= (-1505175+\widehat{L}_0)L_{2n}+5(-673134+\widehat{F}_0)F_{2n}.
\end{equation}
Substituting Eq. (\ref{d2}) into Eq. (\ref{d1}) gives Eq. (\ref{13}). Equation (\ref{12}) can be proved similarly.
\end{proof}

\begin{theorem}
For any integers $n,r$ and $s$, we have
\begin{equation}
\widehat{L}_{n+r}\widehat{F}_{n+s}-\widehat{L}_{n+s}\widehat{F}_{n+r}=2\left(  -1\right)  ^{n+r}\widehat{L}_0  F_{s-r}. 
\end{equation}
\end{theorem}
\begin{proof}
The Binet formulas for the Fibonacci and Lucas sedenions give
\begin{eqnarray*}
\widehat{L}_{n+r}\widehat{F}_{n+s}-\widehat{L}_{n+s}\widehat{F}_{n+r} &amp;=&amp;\frac{1}{\sqrt{5}}\Bigl(  \bigl(
\widehat{\alpha}\alpha^{n+r}+\widehat{\beta}\beta^{n+r}\bigr)  \bigl(  \widehat{\alpha}\alpha^{n+s}
-\widehat{\beta}\beta^{n+s}\bigr)   \\
&amp;&amp;\hspace{0.2cm} -\bigl(
\widehat{\alpha}\alpha^{n+s}+\widehat{\beta}\beta^{n+s}\bigr)  \bigl(  \widehat{\alpha}\alpha^{n+r}
-\widehat{\beta}\beta^{n+r}\bigr) \Bigr)\\
&amp;=&amp;\frac{1}{\sqrt{5}}\bigl(  -\widehat{\alpha}\widehat{\beta}\alpha^{n+r}\beta
^{n+s}+\widehat{\beta}\widehat{\alpha}\beta^{n+r}\alpha^{n+s}\\
&amp;&amp;\hspace{1cm}+\widehat{\alpha}
\widehat{\beta}\alpha^{n+s}\beta^{n+r}-\widehat{\beta}\widehat{\alpha}\alpha^{n+r}
\beta^{n+s}\bigr).
\end{eqnarray*}
Using Eqs. (\ref{7a}) and (\ref{8a}), we have
\begin{eqnarray*}
\widehat{L}_{n+r}\widehat{F}_{n+s}-\widehat{L}_{n+s}\widehat{F}_{n+r}&amp;=&amp;\frac{2}{\sqrt{5}}\bigl(  -\alpha^{n+r}\beta^{n+s}\widehat{L}_0   \\
&amp;&amp;\hspace{2cm}+\alpha^{n+s}\beta^{n+r}\widehat{L}_0  \bigr) \\
&amp;=&amp;\frac{2}{\sqrt{5}}\widehat{L}_0  \Bigl(  \alpha^{n+r}\beta
^{n+r}\left(  \alpha^{s-r}-\beta^{s-r}\right)  \Bigr)  \\
&amp;=&amp;2\left(  -1\right)^{n+r}\widehat{L}_0  F_{s-r}.
\end{eqnarray*}
\end{proof}

Some identities for Fibonacci and Lucas sedenions are given without proof in the next theorem.

\begin{theorem}
For any integers $m$ and $n$, we have
\begin{equation*}
\widehat{F}_{m+n}+(  -1)^{n}\widehat{F}_{m-n}=\widehat{F}_{m}L_{n},
\end{equation*}
\begin{equation*}
\widehat{F}_{m}\widehat{L}_{n}-\widehat{L}_{n}\widehat{F}_{m}=-2( -1) ^{m}\widehat{L}_0F_{n-m}, 
\end{equation*}
\begin{equation*}
\widehat{F}_{m}\widehat{L}_{n}-\widehat{L}_{m}\widehat{F}_{n}=-2(-1)^{m}\Bigl(  \widehat{L}_0  F_{n-m}+(\widehat{F}_0-\omega) L_{n-m}\Bigr), 
\end{equation*}
\begin{equation*}
\widehat{F}_{n}\widehat{F}_{m}-\widehat{F}_{m}\widehat{F}_{n}=-2(-1)  ^{m}(\widehat{F}_0-\omega) F_{n-m}, 
\end{equation*}
and 
\begin{equation*}
\widehat{L}_{n}\widehat{L}_{m}-\widehat{L}_{m}\widehat{L}_{n}=10(-1)^{m}(\widehat{F}_0-\omega) F_{n-m}. 
\end{equation*}
\end{theorem}

The following interesting identities for Fibonacci and Lucas sedenions come directly from
their definitions.

\begin{corollary}
Fibonacci and Lucas sedenions satisfy
\begin{equation*}
\widehat{F}_{n+r}F_{n+r}-\widehat{F}_{n-r}F_{n-r}=\widehat{F}_{2n}F_{2r}, 
\end{equation*}
\begin{equation*}
\widehat{F}_{n+r}L_{n+r}+\widehat{F}_{n-r}L_{n-r}=\widehat{F}_{2n}L_{2r}+2(-1)^{n+r}\widehat{F}_{0},
\end{equation*}
\begin{equation*}
\widehat{L}_{n+r}L_{n+r}-\widehat{L}_{n-r}L_{n-r}=5\widehat{F}_{2n}F_{2r},
\end{equation*}
and
\begin{equation*}
\widehat{L}_{n+r}L_{n+r}+\widehat{L}_{n-r}L_{n-r}=L_{2r}\widehat{L}_{2n}+2(-1)^{n+r}\widehat{L}_{0}.
\end{equation*}
\end{corollary}


\bigskip

\begin{thebibliography}{9}

\bibitem{a_k} I. Akkus and O. Kecilioglu, Split Fibonacci and Lucas octonions, \textit{Adv. Appl. Clifford Algebr.} \textbf{25} (2015), 517--525.

\bibitem{car} A. Cariow and G. Cariowa, An algorithm for fast multiplication of sedenions, \textit{Inform. Process. Lett.} \textbf{113} (2013), 324--331.

\bibitem{caw} R. E. Cawagas, On the structure and zero divisors of the Cayley-Dickson sedenion algebra, \textit{Discuss. Math. Gen. Algebra Appl.} \textbf{24} (2004), 251--265.

\bibitem{hal12} S. Halici, On Fibonacci quaternions, \textit{Adv. Appl. Clifford Algebr.} \textbf{22} (2012), 321--327.

\bibitem{har} C. J. Harman, Complex Fibonacci numbers, \textit{Fibonacci Quart.} \textbf{19} (1981), 82--86.

\bibitem{hor63} A. F. Horadam, Complex Fibonacci Numbers and Fibonacci quaternions, \textit{Amer. Math. Monthly} \textbf{70} (1963), 289--291.

\bibitem{hor93} A. F. Horadam, Quaternion recurrence relations, \textit{Ulam Quarterly} \textbf{2} (1993), 23--33.

\bibitem{ima} K. Imaeda and  M. Imaeda, Sedenions: algebra and analysis, \textit{Appl. Math. Comput.} \textbf{115} (2000), 77--88.

\bibitem{iye} M. R. Iyer, A note on Fibonacci quaternions, \textit{Fibonacci Quart.} \textbf{7} (1969), 225--229.

\bibitem{k-a} O. Kecilioglu and I. Akkus, The Fibonacci octonions, \textit{Adv. Appl. Clifford Algebr.} \textbf{25} (2015), 151--158.

\bibitem{kos} T. Koshy, \textit{Fibonacci and Lucas Numbers with Applications}, A Wiley-Interscience Publication, 2001.

\bibitem{swa} M. N. S. Swamy, On generalized Fibonacci quaternions, \textit{Fibonacci Quart.} \textbf{11} (1973), 547--550.



\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}: Primary 11B39; Secondary 20N05, 17A45. \\
\noindent \emph{Keywords:} sedenion, Fibonacci sedenion, Lucas sedenion.

\bigskip
\hrule
\bigskip
\noindent (Concerned with sequences
\seqnum{A000045} and
\seqnum{A000032}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received July 13 2016;
revised versions received December 1 2016; December 27 2016.
Published in {\it Journal of Integer Sequences}, December 27 2016.

\bigskip
\hrule
\bigskip

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


\end{document}

                                                                                

</pre></body></html>