Elif Tan: On generalized Horadam sequence

Abstract: In this talk we consider a generalization of Horadam sequence, called as biperiodic Horadam sequence $ {w_{n} }$ which is defined by the recurrence relation:

[ w_n = a^{\xi(n+1)} b^{\xi(n)} w_{n-1} + c w_{n-2} ]

with arbitrary initial values $w_{0}, w_{1}$ and non zero real numbers $a,b,c$ . Here $\xi(n) = [1- (-1)^{n}]/2$ , that is $\xi(n)=0$, when n is even and $\xi(n)=0$ when $n$ is odd. This sequence emerges as a natural extension of the classical Horadam sequence when $a=b$. We provide a combinatorial interpretation for the bi-periodic Horadam numbers by using the weighted tilings approach. Additionally, we introduce biperiodic incomplete Horadam numbers and bi-periodic hyper Horadam numbers, and give a relationship between them.

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