Journal of Integer Sequences, Vol. 19 (2016), Article 16.4.6

A q-Analogue of the Bi-Periodic Fibonacci Sequence

José L. Ramírez
Departamento de Matemáticas
Universidad Sergio Arboleda
Calle 74 No. 14 — 14

Víctor F. Sirvent
Departamento de Matemáticas
Universidad Simón Bolívar
Apartado 89000
Caracas 1086-A


The Fibonacci sequence has been generalized in many ways. One of them is defined by the relation tn = atn-1 + tn-2 if n is even, and tn = btn-1 + tn-2 if n is odd, with initial values t0 = 0 and t1 = 1, where a and b are positive integers. This sequence is called the bi-periodic Fibonacci sequence. In the present article, we introduce a q-analog of the bi-periodic Fibonacci sequence, and prove several identities involving this sequence. We also give a combinatorial interpretation of this q-analog bi-periodic Fibonacci sequence in terms of weighted colored tilings.

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(Concerned with sequence A000045.)

Received November 4 2015; revised version received March 30 2016. Published in Journal of Integer Sequences, May 9 2016.

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