@article{article_825908, title={Bi-periodic $r$-Fibonacci sequence and bi-periodic $r$-Lucas sequence of type $s$}, journal={Hacettepe Journal of Mathematics and Statistics}, volume={51}, pages={680–699}, year={2022}, DOI={10.15672/hujms.825908}, author={Aıt-amrane, Nacima Rosa and Belbachir, Hacène}, keywords={bi-periodic r-Fibonacci sequence, bi-periodic r-Lucas sequence, companion sequence, recurrence relation, generating function, explicit formula, Binet form}, abstract={In the present paper, for a positive integer $r$, we study bi-periodic $r$-Fibonacci sequence and its family of companion sequences, bi-periodic $r$-Lucas sequence of type $s$ with $1 \leq s \leq r$, which extend the classical Fibonacci and Lucas sequences. Afterwards, we establish the link between the bi-periodic $r$-Fibonacci sequence and its companion sequences. Furthermore, we give their properties as linear recurrence relations, generating functions, explicit formulas and Binet forms.}, number={3}, publisher={Hacettepe University}