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The Generalized Bi-Periodic Fibonacci Sequence Modulo ***m*

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Hacène Belbachir and Celia Salhi

USTHB

Faculty of Mathematics

RECITS Laboratory

16111 Bab Ezzouar, Algiers

Algeria

**Abstract:**

For given positive integers *a,b,c,* and *d*,
we consider the generalized bi-periodic Fibonacci sequence
(*F*_{n})_{n≥0} defined by the
recurrence relation *F*_{n} =
*aF*_{n−1} + *cF*_{n−2}
for *n* even and *F*_{n} =
*bF*_{n−1} + *dF*_{n−2}
for *n* odd, with initial conditions *F*_{0} =
0, *F*_{1} = 1. In the present paper, we study the
periodicity of (*F*_{n})_{n≥0}
modulo a given integer *m* ≥ 2 relatively prime to *c*
and *d*. We extend some well-known results on the period and the
rank of the classical Fibonacci sequence to the bi-periodic case.

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(Concerned with sequences
A000045
A000129
A002530
A048788.)

Received July 3 2021; revised versions received July 9 2021; September 20 2021; September 21 2021; September 24 2021.
Published in *Journal of Integer Sequences*,
October 1 2021.

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