An identity relating Fibonacci and Lucas
numbers of order k
Spiros D. Dafnis
1
Andreas N. Philippou
2
Department of Mathematics,
University of Patras,
Patras,
GR 265-00, Greece
Ioannis E. Livieris
3
Department of Computer & Informatics Engineering,
Technological Educational Institute of Western Greece,
Antirrio,
GR 263-34, Greece
Abstract
The following relation between Fibonacci and Lucas numbers of order k,
n
∑
i=0
m
i
L
(k)
i
+(m - 2)F
(k)
i+1
-
k
∑
j=3
( j - 2)F
(k)
i- j+1
= m
n+1
F
(k)
n+1
+ k - 2,
is derived by means of colored tiling. This relation generalizes the well-known Fibonacci
- Lucas identities, ∑
n
i=0
2
i
L
i
= 2
n+1
F
n+1
, ∑
n
i=0
3
i
(L
i
+ F
i+1
)= 3
n+1
F
n+1
and ∑
n
i=0
m
i
(L
i
+
(m - 2)F
i+1
)= m
n+1
F
n+1
of A.T. Benjamin and J.J. Quinn, D. Marques, and T. Edgar,
respectively.
Keywords: Fibonacci numbers, Lucas numbers, order k, color tiling, generalization.
1
Email: dafnisspyros@gmail.com
2
Email: anphilip@math.upatras.gr
3
Email: livieris@teiwest.gr
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 70 (2018) 37–42
1571-0653/© 2018 Elsevier B.V. All rights reserved.
www.elsevier.com/locate/endm
https://doi.org/10.1016/j.endm.2018.11.006