Identities Involving the Fourth-Order Linear Recurrence Sequence

Qi, Lan; Chen, Zhuoyu

doi:10.3390/sym11121476

Cited by 2 publications

(2 citation statements)

References 16 publications

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“…In conclusion, g is increasing for x ≥ k + 1. In particular, g(n) > g(k) = 0 and so (16) holds. This finishes the proof.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…The Fibonacci sequence was generalized in many different ways, some of them you can find in [6][7][8][9][10][11][12][13][14][15][16][17][18]. By keeping its order (which is 2), we have a general Lucas sequence (C n ) n = (C n (a, b)) n which is defined by the recurrence C n = aC n−1 + bC n−2 , for n ≥ 2, and with C i = i, for i ∈ {0, 1} (moreover, the integer parameters a and b must be such that if x 2 − ax − b = (x − r)(x − s), then r/s is not a root of unity).…”

Section: Introductionmentioning

confidence: 99%

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Some Diophantine Problems Related to k-Fibonacci Numbers

Trojovský

Hubálovský

2020

Mathematics

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Let k ≥ 1 be an integer and denote ( F k , n ) n as the k-Fibonacci sequence whose terms satisfy the recurrence relation F k , n = k F k , n − 1 + F k , n − 2 , with initial conditions F k , 0 = 0 and F k , 1 = 1 . In the same way, the k-Lucas sequence ( L k , n ) n is defined by satisfying the same recursive relation with initial values L k , 0 = 2 and L k , 1 = k . The sequences ( F k , n ) n ≥ 0 and ( L k , n ) n ≥ 0 were introduced by Falcon and Plaza, who derived many of their properties. In particular, they proved that F k , n 2 + F k , n + 1 2 = F k , 2 n + 1 and F k , n + 1 2 − F k , n − 1 2 = k F k , 2 n , for all k ≥ 1 and n ≥ 0 . In this paper, we shall prove that if k > 1 and F k , n s + F k , n + 1 s ∈ ( F k , m ) m ≥ 1 for infinitely many positive integers n, then s = 2 . Similarly, that if F k , n + 1 s − F k , n − 1 s ∈ ( k F k , m ) m ≥ 1 holds for infinitely many positive integers n, then s = 1 or s = 2 . This generalizes a Marques and Togbé result related to the case k = 1 . Furthermore, we shall solve the Diophantine equations F k , n = L k , m , F k , n = F n , k and L k , n = L n , k .

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“…In conclusion, g is increasing for x ≥ k + 1. In particular, g(n) > g(k) = 0 and so (16) holds. This finishes the proof.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Some Diophantine Problems Related to k-Fibonacci Numbers

Trojovský

Hubálovský

2020

Mathematics

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On Certain Fourth-Order Linear Recursive Sequences

Karadeniz-Gözeri,

Sarı,

Akgül

2024

Symmetry

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In this paper, we introduce a fourth-order linear recursive sequence that is related to the concept of subbalancing numbers. This sequence is constructed by using the third balancing number in the Diophantine equation of subbalancing numbers and is called the sequence of B3-Lucas subbalancing numbers. Motivated by the results for b3-Lucas subbalancing numbers, we obtain several algebraic identities regarding B3-Lucas subbalancing numbers. Furthermore, we derive some algebraic relations between B3-Lucas subbalancing numbers and some other integer sequences.

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Identities Involving the Fourth-Order Linear Recurrence Sequence

Cited by 2 publications

References 16 publications

Some Diophantine Problems Related to k-Fibonacci Numbers

Some Diophantine Problems Related to k-Fibonacci Numbers

On Certain Fourth-Order Linear Recursive Sequences

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