Geodesic
A $q$-analogue of the bi-periodic Fibonacci sequence
Journal of integer sequences, Tome 19 (2016) no. 4.

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Summary: 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.
Keywords: q-Fibonacci sequence, q-bi-periodic Fibonacci sequence, bi-periodic Fibonacci sequence, q-analogues, combinatorial identities 
@article{JIS_2016__19_4_a4,
     author = {Ram{\'\i}rez, Jos\'e L. and Sirvent, V{\'\i}ctor F.},
     title = {A $q$-analogue of the bi-periodic {Fibonacci} sequence},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {19},
     number = {4},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2016__19_4_a4/}
}
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Ramírez, José L.; Sirvent, Víctor F. A $q$-analogue of the bi-periodic Fibonacci sequence. Journal of integer sequences, Tome 19 (2016) no. 4. http://geodesic.mathdoc.fr/item/JIS_2016__19_4_a4/