On $k$-generalized Fibonacci numbers with negative indices

Pethö, Attila

doi:10.5486/pmd.2021.8912

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2022

2023

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Cited by 4 publications

(2 citation statements)

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“…in integers n, m, where (T n ) n≥0 denotes the Tribonacci sequence, which is defined by the initial terms T 0 = T 1 = 0, T 1 = 1 and by the recursion T n+3 = T n+2 + T n+1 + T n for n ≥ 0. Pethő [13] generalized the results of Bravo et al [3,4] to generalized Fibonacci numbers, showing that equation…”

Section: Introductionmentioning

confidence: 88%

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Common Values of Padovan and Perrin Sequences

Bravo

2023

Mediterr. J. Math.

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The integer sequence defined by $$P_{n+3}=P_{n+1}+P_{n}$$ P n + 3 = P n + 1 + P n with initial conditions $$P_{0}=P_{1}=P_{2}=1$$ P 0 = P 1 = P 2 = 1 is known as the Padovan sequence $$(P_{n})_{n\in \mathbb {Z}}$$ ( P n ) n ∈ Z . The Perrin sequence $$(R_{m})_{m\in \mathbb {Z}}$$ ( R m ) m ∈ Z satisfies the same recurrence equation as the Padovan sequence but with starting values $$R_{0}=3$$ R 0 = 3 , $$R_{1}=0$$ R 1 = 0 , and $$R_{2}=2$$ R 2 = 2 . In this note, we solve the Diophantine equation $$P_{n}=\pm R_{m}$$ P n = ± R m with $$(n,m)\in \mathbb {Z}^{2}$$ ( n , m ) ∈ Z 2 .

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Section: Introductionmentioning

confidence: 88%

“…Of course, for k = 3, we get the Tribonacci sequence. Unfortunately, the proof of Pethő [13] is ineffective, since it is based on the theory of S-unit equations. See also Pethő and Szalay [14].…”

Section: Introductionmentioning

confidence: 99%