Natural density and probability, constructively

Maschio, Samuele

doi:10.4467/20842589rm.20.002.12434

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2021

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“…Clearly, this is not a surprising fact, given the exponential nature of Fibonacci sequence. We refer the reader to the classical work of Niven [1] (and references therein) for more details about natural (and asymptotic) density (see also books [2,3] and papers [4][5][6][7][8] for more recent results).…”

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confidence: 99%

On the Natural Density of Sets Related to Generalized Fibonacci Numbers of Order r

Trojovský

2021

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For r≥2 and a≥1 integers, let (tn(r,a))n≥1 be the sequence of the (r,a)-generalized Fibonacci numbers which is defined by the recurrence tn(r,a)=tn−1(r,a)+⋯+tn−r(r,a) for n>r, with initial values ti(r,a)=1, for all i∈[1,r−1] and tr(r,a)=a. In this paper, we shall prove (in particular) that, for any given r≥2, there exists a positive proportion of positive integers which can not be written as tn(r,a) for any (n,a)∈Z≥r+2×Z≥1.

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