A Note on Bi-Periodic Leonardo Sequence

Abstract

In this work, we define a new generalization of the Leonardo sequence by the recurrence relation $GLe_n=aGLe_{n-1}+GLe_{n-2}+a$ (for even $n$) and $GLe_n=bGLe_{n-1}+GLe_{n-2}+b$ (for odd $n$) with the initial conditions $GLe_0=2a-1$ and $GLe_1=2ab-1$, where $a$ and $b$ are real nonzero numbers. Some algebraic properties of the sequence $\{GLe_n\}_{n \geq 0}$ are studied and several identities, including the generating function and Binet's formula, are established.