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arXiv · 1508.02762

A Family of the Zeckendorf Theorem Related Identities

Ivica Martinjak
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Abstract

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.

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Ivica Martinjak. 2015-08-11. A Family of the Zeckendorf Theorem Related Identities. https://arxiv.org/abs/1508.02762

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