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A ***q*-Analogue of the Bi-Periodic Fibonacci Sequence

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José L. Ramírez

Departamento de Matemáticas

Universidad Sergio Arboleda

Calle 74 No. 14 — 14

Bogotá

Colombia

Víctor F. Sirvent

Departamento de Matemáticas

Universidad Simón Bolívar

Apartado 89000

Caracas 1086-A

Venezuela

**Abstract:**

The Fibonacci sequence has been generalized in many ways. One of them
is defined by the relation
*t*_{n} = *at*_{n-1} + *t*_{n-2} if *n* is even,
and
*t*_{n} = *bt*_{n-1} + *t*_{n-2}
if *n* is odd, with initial values *t*_{0} = 0 and
*t*_{1} = 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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