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

Department of Mathematics

University of Wisconsin-Eau Claire

Eau Claire, WI 54702

USA

**Abstract:**

We explore the sums of *k* consecutive terms in the generalized Fibonacci sequence
given by the recurrence
*G*_{n} = *G*_{n-1} + *G*_{n-2} for all
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
.
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
,
we give two tantalizing characterizations for these values, one involving a simple formula in *k* and another involving generalized Pisano periods:

where 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.

where denotes the generalized Pisano period of the generalized Fibonacci sequence modulo

(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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