On the connections between Fibonacci and Mulatu Numbers

Eudes Antonio Costa; Élis Gardel da Costa Mesquita; Paula M. M. C. Catarino

doi:10.22481/intermaths.v6i1.16742

Authors

Eudes Antonio Costa Universidade Federal do Tocantins, Arraias, Tocantins, BR https://orcid.org/0000-0001-6684-9961
Élis Gardel da Costa Mesquita Universidade Federal do Tocantins, Palmas, Tocantins, BR https://orcid.org/0000-0003-2385-4108
Paula M. M. C. Catarino Universidade de Trás-os-Montes e Alto Douro, Vila Real, Vila Real, PT https://orcid.org/0000-0001-6917-5093

DOI:

https://doi.org/10.22481/intermaths.v6i1.16742

Keywords:

Generating function , Fibonacci numbers, Fibonacci-type numbers, Golden ratio

Abstract

In this work, we present a detailed study of the Fibonacci--Mulatu sequence, {FM_n}, defined recursively by FM_n+2=FM_n+1+FM_n with initial terms FM₀ = 4 and FM₁ = 1. We establish several properties of this sequence, elucidating the connections it evinces with the classical Fibonacci and Fibonacci--Lucas sequences. Additionally, we derive an expression for negative indexes. Specifically, we derive the Binet formula for this sequence and determine the limit for quotients involving both positive and negative indexes. Additionally, we construct three types of generating functions--ordinary, exponential, and Poisson--and examine classical identities, including those of d'Ocagne, Catalan, Cassini, Melham, Cesàro, as well as the convolution identity. We further calculate partial sums and alternating partial sums of this sequence.

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