Narayana Numbers With Zeckendorf Partition in Two Terms

Japhet Odjoumani and Salifou Nikiema

International Journal of Mathematics and Mathematical Sciences, 2025, vol. 2025, 1-7

Abstract: The Narayan’s cow sequence starts with the terms 1, 1, and 1. Each subsequent term is obtained as the sum of the previous term and the term three places before. A term of this sequence is called a Narayana number. The mathematician Zeckendorf proved that every positive integer has a unique decomposition into a sum of distinct and nonconsecutive Fibonacci numbers, known as the Zeckendorf partition. In this paper, we determine all Narayana numbers that can be expressed as the sum of two Fibonacci numbers. We show that these numbers are precisely the fourth, fifth, sixth, seventh, eighth, ninth, and thirteenth Narayana numbers. In particular, we prove that the only Narayana numbers whose Zeckendorf partition contains exactly two terms are the sixth, seventh, eighth, and thirteenth Narayana numbers.

Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:6633605

DOI: 10.1155/ijmm/6633605

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