2015/08/11 by Ivica Martinjak, Martinjak, Ivica
Mathematics · #11B37 #11B39 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:11B37 #msc:11B39
paper · pdf · doi:10.48550/arxiv.1508.02762
10 pages
arxiv created 2015/08/11 · arxiv updated 2015/08/13
In this paper we present a family of identities for recursive sequences arising from a second order recurrence relation, that gives instances of Zeckendorf representation. We prove these results using a special case of an universal property of the recursive sequences. In particular cases we also establish a direct bijection. Besides, we prove further equalities that provide a representation of the sum of (r+1)-st and (r-1)-st Fibonacci number as the sum of powers of the golden ratio. Similarly, we show a class of natural numbers represented as the sum of powers of the silver ratio.