Fibonacci and Lucas Sedenions

Göksal Bilgici

Department of Computer Education and Instructional Technology

Education Faculty Kastamonu University Kastamonu, 37100 Turkey

               gbilgici@kastamonu.edu.tr

Ümit Tokeşer and Zafer Ünal Department of Mathematics Faculty of Arts and Sciences Kastamonu University Kastamonu, 37150 Turkey

               utokeser@kastamonu.edu.tr
                zunal@kastamonu.edu.tr

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.

1 Introduction

Sedenions appear in many areas of science, such as electromagnetic theory and linear gravity. Sedenion algebra, which is usually denoted by 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 𝕊 as {e_{0}, e_{1}, …, e_{15}}, where e_{0} is the unit element and e_{1}, …, e_{15} are imaginaries. A sedenion S can be written as

S = ∑_{i=0}^{15} a_{i}e_{i}

where a_{0}, a_{1}, ⋯, a_{15} are reals.

Imaeda and Imaeda [8] defined a sedenion by

S = (O_{1}; O_{2}) ∈ 𝕊, O_{1}, O_{2} ∈ 𝕆

where 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_{1}S_{2} = (O_{1}O_{3} + ρ\overline{O_{4}}O_{2}; O_{2}\overline{O_{3}} + O_{4}O_{1}).

After choosing the field parameter ρ = −1 and the generator e_{8}, Imaeda and Imaeda examined the sedenions. By setting i ≡ e_{i}, where i = 0, 1, ⋯, 15, Cawagas [3] constructed the following multiplication table for the basis of 𝕊.

\begin{matrix}· & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ 1 & 1 & −0 & 3 & −2 & 5 & −4 & −7 & 6 & 9 & −8 & −11 & 10 & −13 & 12 & 15 & −14 \\ 2 & 2 & −3 & −0 & 1 & 6 & 7 & −4 & −5 & 10 & 11 & −8 & −9 & −14 & −15 & 12 & 13 \\ 3 & 3 & 2 & −1 & −0 & 7 & −6 & 5 & −4 & 11 & −10 & 9 & −8 & −15 & 14 & −13 & 12 \\ 4 & 4 & −5 & −6 & −7 & −0 & 1 & 2 & 3 & 12 & 13 & 14 & 15 & −8 & −9 & −10 & −11 \\ 5 & 5 & 4 & −7 & 6 & −1 & −0 & −3 & 2 & 13 & −12 & 15 & −14 & 9 & −8 & 11 & −10 \\ 6 & 6 & 7 & 4 & −5 & −2 & 3 & −0 & −1 & 14 & −15 & −12 & 13 & 10 & −11 & −8 & 9 \\ 7 & 7 & −6 & 5 & 4 & −3 & −2 & 1 & −0 & 15 & 14 & −13 & −12 & 11 & 10 & −9 & −8 \\ 8 & 8 & −9 & −10 & −11 & −12 & −13 & −14 & −15 & −0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 9 & 9 & 8 & −11 & 10 & −13 & 12 & 15 & −14 & −1 & −0 & −3 & 2 & −5 & 4 & 7 & −6 \\ 10 & 10 & 11 & 8 & −9 & −14 & −15 & 12 & 13 & −2 & 3 & −0 & −1 & −6 & −7 & 4 & 5 \\ 11 & 11 & −10 & 9 & 8 & −15 & 14 & −13 & 12 & −3 & −2 & 1 & −0 & −7 & 6 & −5 & 4 \\ 12 & 12 & 13 & 14 & 15 & 8 & −9 & −10 & −11 & −4 & 5 & 6 & 7 & −0 & −1 & −2 & −3 \\ 13 & 13 & −12 & 15 & −14 & 9 & 8 & 11 & −10 & −5 & −4 & 7 & −6 & 1 & −0 & 3 & −2 \\ 14 & 14 & −15 & −12 & 13 & 10 & −11 & 8 & 9 & −6 & −7 & −4 & 5 & 2 & −3 & −0 & 1 \\ 15 & 15 & 14 & −13 & −12 & 11 & 10 & −9 & 8 & −7 & 6 & −5 & −4 & 3 & 2 & −1 & −0 \end{matrix}

Table 1: Multiplication table for the basis of 𝕊

Cariow and Cariowa [2] 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 ≥ 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 (F_{n})_{n≥0} and Lucas sequence (L_{n})_{n≥0} are

∑_{n=0}^{∞} F_{n}x^{n} = \frac{x}{1 − x − x^{2}} and ∑_{n=0}^{∞} L_{n}x^{n} = \frac{2 − x}{1 − x − x^{2}} .

The Binet formulas for the Fibonacci and Lucas numbers are

F_{n} = \frac{α^{n} − β^{n}}{α − β} and L_{n} = α^{n} + β^{n}  (1)

where α = \frac{1+\sqrt{5}}{2} and β = \frac{1−\sqrt{5}}{2} are the roots of the characteristic equation of x^{2} − x − 1 = 0. The positive root α is known as the golden ratio (see [11] for details).

Horadam [6] 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 nth classical Fibonacci number and L_{n} is the nth 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 nth 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 [9] described various properties of the Fibonacci quaternions and generalized Fibonacci quaternions. Swammy [12] also studied these two types of quaternions, and obtained some relations between Fibonacci and Lucas numbers. Harman [5] defined complex Fibonacci numbers by the following twodimensional recurrence relation

G(n + 2, m) = G(n + 1, m) + G(n, m),
G(n, m + 2) = G(n, m + 1) + G(n, m)

where G(0, 0) = 0, G(1, 0) = 1, G(0, 1) = i, and G(1, 1) = 1 + i. By extending this idea, Horadam [7] defined the recurrence relations

G(h + 2, l, m, n) = G(h + 1, l, m, n) + G(h, l, m, n),
G(h, l + 2, m, n) = G(h, l + 1, m, n) + G(h, l, m, n),
G(h, l, m + 2, n) = G(h, l, m + 1, n) + G(h, l, m, n),
G(h, l, m, n + 2) = G(h, l, m, n + 1) + G(h, l, m, n)

with initial conditions

G(0, 0, 0, 0) = 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) = 1 + e_{1}, …, G(0, 0, 1, 1) = e_{2} + e_{3},
G(1, 1, 1, 0) = 1 + e_{1} + e_{2}, …, G(0, 1, 1, 1) = e_{1} + e_{2} + e_{3},
G(1, 1, 1, 1) = 1 + e_{1} + e_{2} + e_{3}

and examined quaternion recurrence relations. Halici [4] investigated some properties of the Fibonacci and Lucas quaternions, obtaining generating functions, Binet formulas, and certain identities.

Kecilioglu and Akkus [10] introduced the Fibonacci and Lucas octonions. These hypercomplex numbers are defined as

Q_{n} = ∑_{s=0}^{7} F_{n+s}e_{s}

and

T_{n} = ∑_{s=0}^{7} L_{n+s}e_{s}

respectively, where F_{n} and L_{n} are the nth classical Fibonacci and Lucas numbers. In another study [1], 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 𝕊. The nth Fibonacci sedenion is

F̂_{n} = ∑_{s=0}^{15} F_{n+s}e_{s}  (2)

and the nth Lucas sedenion is

L̂_{n} = ∑_{s=0}^{15} L_{n+s}e_{s}.  (3)

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

F̂_{−n} = ∑_{s=0}^{15}(−1)^{n+s+1}F_{n−s}e_{s}

and

L̂_{−n} = ∑_{s=0}^{15}(−1)^{n+s}L_{n−s}e_{s}.

For all integers n, we can easily see that

F̂_{n} = F̂_{n−1} + F̂_{n−2} and L̂_{n} = L̂_{n−1} + L̂_{n−2}.  (4)

Additionally, for any integer n, we have

L̂_{n} = F̂_{n−1} + F̂_{n+1}.

2 Generating functions and Binet formulas for the Fi-

bonacci and Lucas sedenions

Generating functions for the Fibonacci and Lucas sedenions are given in the next theorem. Theorem 1. The generating functions for the Fibonacci and Lucas sedenions are

∑_{i=0}^{∞} F̂_{i}x^{i} = \frac{F̂_{0} + F̂_{−1}x}{1 − x − x^{2}}  (5)

and

∑_{i=0}^{∞} L̂_{i}x^{i} = \frac{L̂_{0} + L̂_{−1}x}{1 − x − x^{2}}  (6)

respectively. Proof. Define f(x) = ∑_{i=0}^{∞} F̂_{i}x^{i}. Expanding the first two terms of f(x), we have

f(x) = F̂_{0} + F̂_{1}x + ∑_{i=2}^{∞} F̂_{i}x^{i}.  (7)

Multiplying both sides of this equation by −x and −x^{2}, we obtain

−xf(x) = −F̂_{0}x − ∑_{i=2}^{∞} F̂_{i−1}x^{i}  (8)

and

−x^{2}f(x) = − ∑_{i=2}^{∞} F̂_{i−2}x^{i}.  (9)

Adding Eqs. (7), (8) and (9) with consideration of the first equation in Eq. (4) gives

f(x) = \frac{F̂_{0} + (F̂_{1} − F̂_{0})x}{1 − x − x^{2}}.

Using the identity F̂_{−1} = F̂_{1} − F̂_{0}, we have Eq. (5). Similarly, we can obtain Eq. (6).

The next theorem gives the Binet formulas for the Fibonacci and Lucas sedenions. Theorem 2. For any integer n, the nth Fibonacci sedenion is

F̂_{n} = \frac{α̂α^{n} − β̂β^{n}}{α − β}  (10)

and the nth Lucas sedenion is

L̂_{n} = α̂α^{n} + β̂β^{n}  (11)

where α̂ = ∑_{s=0}^{15} α^{s}e_{s} and β̂ = ∑_{s=0}^{15} β^{s}e_{s}.

Proof. For the first equation, we have

αF̂_{n} + F̂_{n−1} = ∑_{s=0}^{15} (αF_{n+s} + F_{n+s−1}) e_{s}.

From the identity αF_{n} + F_{n−1} = α^{n}, we obtain

αF̂_{n} + F̂_{n−1} = α̂α^{n}.  (12)

Similarly, we have

βF̂_{n} + F̂_{n−1} = β̂β^{n}.  (13)

Subtracting Eq. (13) from Eq. (12) gives

(α − β)F̂_{n} = α̂α^{n} − β̂β^{n}

from which we obtain Eq. (10). Summing Eqs. (12) and (13), we have

F̂_{n} + 2F̂_{n−1} = α̂α^{n} + β̂β^{n}

and, with the help of the identity L_{n} = F_{n} + 2F_{n−1}, we obtain Eq. (11).

When using the Binet formulas to obtain identities for the Fibonacci and Lucas sedenions, we require α̂β̂, β̂α̂, α̂^{2}, and β̂^{2}. These products are given in the next lemma. Lemma 3. We have

α̂β̂ = L̂_{0} − \sqrt{5}(F̂_{0} − ω),  (14)
β̂α̂ = L̂_{0} + \sqrt{5}(F̂_{0} − ω),  (15)
α̂^{2} = −1505175 + L̂_{0} + \sqrt{5}(−673134 + F̂_{0}),  (16)

and

β̂^{2} = −1505175 + L̂_{0} − \sqrt{5}(−673134 + F̂_{0})  (17)

where

ω = 94e_{9} + 94e_{10} + 188e_{11} + 282e_{12} − 188e_{13} + 94e_{14} + 893e_{15}.

Proof. From the definitions of α̂ and β̂, and using Table 1, we have

α̂β̂ = (∑_{n=0}^{15} α^{n}e_{n}) (∑_{n=0}^{15} β^{n}e_{n})
= 2 + e_{1} + 3e_{2} + ⋯ + 1364e_{15} − \sqrt{5}(e_{1} + e_{2} + 2e_{3} + ⋯ + 21e_{8}

−60e_{9} − 39e_{10} − 99e_{11} − 138e_{12} + 421e_{13} + 283e_{14} − 283e_{15})

where the final equation gives Eq. (14). The others can be computed similarly.

This lemma gives us the following useful identity:

α̂β̂ + β̂α̂ = 2L̂_{0}.  (18)

3 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.

Theorem 4. For any integers m and n, we have

F̂_{m+n}F̂_{m−n} − F̂_{m}^{2} = (−1)^{m}F_{−n}(F_{n}L̂_{0} − L_{n}(F̂_{0} − ω))  (19)

and

L̂_{m+n}L̂_{m−n} − L̂_{m}^{2} = −5(−1)^{m}F_{−n}(F_{n}L̂_{0} − L_{n}(F̂_{0} − ω)).  (20)

Proof. From the Binet formula for Fibonacci sedenions, we have

F̂_{m+n}F̂_{m−n} − F̂_{m}^{2} = \frac{1}{5}((α̂α^{m+n} − β̂β^{m+n})(α̂α^{m−n} − β̂β^{m−n})
−(α̂α^{m} − β̂β^{m})^{2})
= \frac{1}{5}((−1)^{m−n+1}(α̂β̂α^{2n} + β̂α̂β^{2n})
+(−1)^{m}2L̂_{0}).

We require Eqs. (14) and (15). Using this equation, we obtain

F̂_{m+n}F̂_{m−n} − F̂_{m}^{2} = \frac{1}{5}((−1)^{m−n+1}((L̂_{0} − \sqrt{5}(F̂_{0} − ω))α^{2n}
+(L̂_{0} + \sqrt{5}(F̂_{0} − ω))β^{2n}) + 2(−1)^{m}L̂_{0})
= \frac{1}{5}((−1)^{m−n+1}(L̂_{0}L_{2n} − 5(F̂_{0} − ω)F_{2n})
+2(−1)^{m}L̂_{0})
= \frac{1}{5}(−1)^{m−n+1}L̂_{0}(L_{2n} − 2(−1)^{n})
−(−1)^{m−n+1}F_{2n}(F̂_{0} − ω).

Using the identity 5F_{n}^{2} = L_{2n} − 2(−1)^{n} gives

F̂_{m+n}F̂_{m−n} − F̂_{m}^{2} = (−1)^{m+n+1}(F_{n}^{2}L̂_{0} − F_{2n}(F̂_{0} − ω)).

With the help of the identity F_{2n} = F_{n}L_{n}, we have Eq. (19). Similarly, we can obtain Eq. (20).

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.

Corollary 5. For any integer m, we have

F̂_{m+1}F̂_{m−1} − F̂_{m}^{2} = (−1)^{m}(2F̂_{−1} + ω)  (21)

and

L̂_{m+1}L̂_{m−1} − L̂_{m}^{2} = −5(−1)^{m}(2F̂_{−1} + ω).  (22)

The following theorem gives d’Ocagne’s identities for Fibonacci and Lucas sedenions.

Theorem 6. For any integers n and m, we have

F̂_{m}F̂_{n+1} − F̂_{m+1}F̂_{n} = (−1)^{n}(F_{m−n}L̂_{0} + L_{m−n}(F̂_{0} − ω))

and

L̂_{m}L̂_{n+1} − L̂_{m+1}L̂_{n} = −5(−1)^{n}(F_{m−n}L̂_{0} + L_{m−n}(F̂_{0} − ω)).

Proof. Using the Binet formula for the Fibonacci sedenions gives

F̂_{m}F̂_{n+1} − F̂_{m+1}F̂_{n} = \frac{1}{5}(α̂α^{m} − β̂β^{m})(α̂α^{n+1} − β̂β^{n+1})

−\frac{1}{5}(α̂α^{m+1} − β̂β^{m+1})(α̂α^{n} − β̂β^{n})

= \frac{\sqrt{5}}{5}(−1)^{n}(α̂β̂α^{m−n} − β̂α̂β^{m−n}).

We require Eqs. (14) and (15). Substituting these into the previous equation, we have

F̂_{m}F̂_{n+1} − F̂_{m+1}F̂_{n} = \frac{\sqrt{5}}{5}(−1)^{n}((L̂_{0} − \sqrt{5}(F̂_{0} − ω))α^{m−n}
−(L̂_{0} + \sqrt{5}(F̂_{0} − ω))β^{m−n})
= (−1)^{n}(F_{m−n}L̂_{0} − L_{m−n}(F̂_{0} − ω)).

The second identity in the theorem, i.e., d’Ocagne’s identity for the Lucas sedenions, can be proved similarly.

After deriving these three famous identities, we present some other identities for the Fibonacci and Lucas sedenions.

Theorem 7. For any integer n, we have

L̂_{n}^{2} − F̂_{n}^{2} = \frac{4}{5}(−1505175 + L̂_{0})L_{2n} + 4(−673134 + F̂_{0})F_{2n} + \frac{12}{5}(−1)^{n}L̂_{0}  (23)

and

L̂_{n}^{2} + F̂_{n}^{2} = \frac{6}{5}(−1505175 + L̂_{0})L_{2n} + 6(−673134 + F̂_{0})F_{2n} + \frac{8}{5}(−1)^{n}L̂_{0}.  (24)

Proof. Using the Binet formulas for the Fibonacci and Lucas sedenions, we obtain

L̂_{n}^{2} − F̂_{n}^{2} = (α̂α^{n} + β̂β^{n})^{2} − \frac{1}{5}(α̂α^{n} − β̂β^{n})^{2}
= (α̂^{2}α^{2n} + β̂^{2}β^{2n} + α̂β̂α^{n}β^{n} + β̂α̂α^{n}β^{n})

−\frac{1}{5}(α̂^{2}α^{2n} + β̂^{2}β^{2n} − α̂β̂α^{n}β^{n} − β̂α̂α^{n}β^{n}).

Substituting Eqs. (14) and (15) into the last equation, we have

L̂_{n}^{2} − F̂_{n}^{2} = \frac{4}{5}(α̂^{2}α^{2n} + β̂^{2}β^{2n}) + \frac{12}{5}(−1)^{n}L̂_{0}.  (25)

Using Eqs. (16) and (17), we obtain

(α̂^{2}α^{2n} + β̂^{2}β^{2n}) = (−1505175 + L̂_{0})L_{2n} + 5(−673134 + F̂_{0})F_{2n}.  (26)

Substituting Eq. (26) into Eq. (25) gives Eq. (23). Equation (24) can be proved similarly. Theorem 8. For any integers n, r and s, we have

L̂_{n+r}F̂_{n+s} − L̂_{n+s}F̂_{n+r} = 2 (−1)^{n+r} L̂_{0}F_{s−r}.  (27)

Proof. The Binet formulas for the Fibonacci and Lucas sedenions give

L̂_{n+r}F̂_{n+s} − L̂_{n+s}F̂_{n+r} = \frac{1}{\sqrt{5}}((α̂α^{n+r} + β̂β^{n+r})(α̂α^{n+s} − β̂β^{n+s})
−(α̂α^{n+s} + β̂β^{n+s})(α̂α^{n+r} − β̂β^{n+r}))
= \frac{1}{\sqrt{5}}(−α̂β̂α^{n+r}β^{n+s} + β̂α̂β^{n+r}α^{n+s}
+α̂β̂α^{n+s}β^{n+r} − β̂α̂α^{n+r}β^{n+s}).

Using Eqs. (14) and (15), we have

L̂_{n+r}F̂_{n+s} − L̂_{n+s}F̂_{n+r} = \frac{2}{\sqrt{5}}(−α^{n+r}β^{n+s}L̂_{0}
+α^{n+s}β^{n+r}L̂_{0})
= \frac{2}{\sqrt{5}}L̂_{0}(α^{n+r}β^{n+r} (α^{s−r} − β^{s−r}))
= 2 (−1)^{n+r} L̂_{0}F_{s−r}.

Some identities for Fibonacci and Lucas sedenions are given without proof in the next theorem. Theorem 9. For any integers m and n, we have

F̂_{m+n} + (−1)^{n}F̂_{m−n} = F̂_{m}L_{n},

F̂_{m}L̂_{n} − L̂_{n}F̂_{m} = −2(−1)^{m}L̂_{0}F_{n−m},
F̂_{m}L̂_{n} − L̂_{m}F̂_{n} = −2(−1)^{m}(L̂_{0}F_{n−m} + (F̂_{0} − ω)L_{n−m}),
F̂_{n}F̂_{m} − F̂_{m}F̂_{n} = −2(−1)^{m}(F̂_{0} − ω)F_{n−m},

and

L̂_{n}L̂_{m} − L̂_{m}L̂_{n} = 10(−1)^{m}(F̂_{0} − ω)F_{n−m}.

The following interesting identities for Fibonacci and Lucas sedenions come directly from their definitions. Corollary 10. Fibonacci and Lucas sedenions satisfy

F̂_{n+r}F_{n+r} − F̂_{n−r}F_{n−r} = F̂_{2n}F_{2r},

F̂_{n+r}L_{n+r} + F̂_{n−r}L_{n−r} = F̂_{2n}L_{2r} + 2(−1)^{n+r}F̂_{0},
L̂_{n+r}L_{n+r} − L̂_{n−r}L_{n−r} = 5F̂_{2n}F_{2r},

and

L̂_{n+r}L_{n+r} + L̂_{n−r}L_{n−r} = L_{2r}L̂_{2n} + 2(−1)^{n+r}L̂_{0}.

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[11] T. Koshy, Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, 2001.

[12] M. N. S. Swamy, On generalized Fibonacci quaternions, Fibonacci Quart. 11 (1973), 547–550.

2010 Mathematics Subject Classification: Primary 11B39; Secondary 20N05, 17A45. Keywords: sedenion, Fibonacci sedenion, Lucas sedenion.

(Concerned with sequences \underline{A000045} and \underline{A000032}.)

Received July 13 2016; revised versions received December 1 2016; December 27 2016. Published in Journal of Integer Sequences, December 27 2016.

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