A New Generalization of Leonardo Sequences: Biperiodic Leonardo Sequence - Hasan Gökbaş (Journal of Mathematics)
In this study, we define a new type of number sequence called biperiodic Leonardo sequence by the recurrence relation Lena,b=aLen−1+Len−2+1 (for even n) and Lena,b=bLen−1+Len−2+1 (for odd n) with the initial conditions Le0a,b=Le1a,b=1. We obtained the characteristic function, generating function, and Binet’s formula for this sequence and propose a determinantal representation for the generating function of this sequence. We also provide some properties of these numbers. Moreover, we give the matrix representation of the biperiodic Leonardo numbers.