An Interesting Generalization of Fibonacci & Lucas Sequence

Vandana R. Patel, Devbhadra V. Shah

Abstract: In this paper, we consider the generalisation of classical Fibonacci sequence and Lucas sequence. We consider the sequence{H_n }defined by the recurrence relation H_n= H_ (n-1) +H_ (n-2), for all n-2, with H_0=2m, H_1=k+m, where m, k are fixed integers. The initial conditions are the sum ofk times the initial conditions of Fibonacci sequence and m times the initial conditions of Lucas sequence. Using the technique of generating functions, we obtain the extended Binet formula for H_n. We obtain some fascinating properties for this sequence. We also establish some amusing identities for this sequence displaying the relation betweenH_n, Fibonacci sequence and Lucas sequence

Keywords: Fibonacci sequence, Lucas sequence, generating function, Generalized Fibonacci sequence