An Interesting Generalization of Fibonacci & Lucas Sequence
Journal: International Journal of Science and Research (IJSR) (Vol.4, No. 12)Publication Date: 2015-12-05
Authors : Vandana R. Patel; Devbhadra V. Shah;
Page : 1942-1945
Keywords : Fibonacci sequence; Lucas sequence; generating function; Generalized Fibonacci sequence;
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
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