GCD of sums of $k$ consecutive Fibonacci, Lucas, and generalized Fibonacci numbers

Abstract

We explore the sums of k consecutive terms in the generalized Fibonacci sequence {G n } ∞ n=0 given by the recurrence G n = G n−1 + G n−2 for all n ≥ 2 with integral initial conditions G 0 and G 1 . In particular, we give precise values for the greatest common divisor (GCD) of all sums of k consecutive terms of {G n } ∞ n=0 . When G 0 = 0 and G 1 = 1, we yield the GCD of all sums of k consecutive Fibonacci numbers, and when G 0 = 2 and G 1 = 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 G G0,G1 (k), we give two tantalizing characterizations for these values, one involving a simple formula in k and another involving generalized Pisano periods:where π G0,G1 (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.