  We explore the sums of $k$ consecutive terms in the generalized Fibonacci
sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} +
G_{n-2}$ for all $n \geq 2$ 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 $\left(G_n\right)_{n \geq 0}$. 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 $\mathcal{G}_{G_0, G_1}\!(k)$, we
give two tantalizing characterizations for these values, one involving a simple
formula in $k$ and another involving generalized Pisano periods:
$$\mathcal{G}_{G_0, G_1}\!(k) = \gcd(G_{k+1}-G_1,\, G_{k+2}-G_2)\; \mbox{and}$$
$$\mathcal{G}_{G_0, G_1}\!(k) = \mathrm{lcm}\{m \mid \pi_{G_0,G_1}\!(m) \text{
divides } k\},$$ 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.
