Journal of Integer Sequences, Vol. 24 (2021), Article 21.9.4

The Generalized Bi-Periodic Fibonacci Sequence Modulo m

Hacène Belbachir and Celia Salhi
Faculty of Mathematics
RECITS Laboratory
16111 Bab Ezzouar, Algiers


For given positive integers a,b,c, and d, we consider the generalized bi-periodic Fibonacci sequence (Fn)n≥0 defined by the recurrence relation Fn = aFn−1 + cFn−2 for n even and Fn = bFn−1 + dFn−2 for n odd, with initial conditions F0 = 0, F1 = 1. In the present paper, we study the periodicity of (Fn)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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