Journal of Integer Sequences, Vol. 24 (2021), Article 21.9.8

GCD of Sums of k Consecutive Fibonacci, Lucas, and Generalized Fibonacci Numbers

Dan Guyer and aBa Mbirika
Department of Mathematics
University of Wisconsin-Eau Claire
Eau Claire, WI 54702


We explore the sums of k consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence Gn = Gn-1 + Gn-2 for all $n \geq 2$ with integral initial conditions G0 and G1. In particular, we give precise values for the greatest common divisor (GCD) of all sums of k consecutive terms of $\left(G_n\right)_{n \geq 0}$. When G0 = 0 and G1 = 1, we yield the GCD of all sums of k consecutive Fibonacci numbers, and when G0 = 2 and G1 = 1, we yield the GCD of all sums of k consecutive Lucas numbers. Denoting the GCD of all sums of k consecutive generalized Fibonacci numbers by the symbol $\mathcal{G}_{G_0, G_1}\!(k)$, we give two tantalizing characterizations for these values, one involving a simple formula in k and another involving generalized Pisano periods:
\begin{align*}\mathcal{G}_{G_0, G_1}\!(k) &= \gcd(G_{k+1}-G_1,\, G_{k+2}-G_2)\; ...
...k) &= \mathrm{lcm}\{m \mid \pi_{G_0,G_1}\!(m) \text{ divides } k\},

where $\pi_{G_0,G_1}\!(m)$ denotes the generalized Pisano period of the generalized Fibonacci sequence modulo m. The fact that these vastly different-looking formulas coincide leads to some surprising and delightful new understandings of the Fibonacci and Lucas numbers.

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(Concerned with sequences A210209 A229339.)

Received May 25 2021; revised version received November 3 2021. Published in Journal of Integer Sequences, November 3 2021.

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