We explore the sums of
k consecutive terms in the generalized Fibonacci sequence
![$\left(G_n\right)_{n \geq 0}$](abs/img1.gif)
given by the recurrence
Gn =
Gn-1 +
Gn-2 for all
![$n \geq 2$](abs/img2.gif)
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
![$\left(G_n\right)_{n \geq 0}$](abs/img1.gif)
.
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
![$\mathcal{G}_{G_0, G_1}\!(k)$](abs/img3.gif)
,
we give two tantalizing characterizations for these values, one involving a simple formula in
k and another involving generalized Pisano periods:
where
![$\pi_{G_0,G_1}\!(m)$](abs/img5.gif)
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.