Sums of Fibonacci–Pell–Jacobsthal products

Koshy, Thomas

doi:10.1080/0020739x.2012.729679

Cited by 3 publications

(3 citation statements)

References 3 publications

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“…For more information see, [2], [4], [6], and [9]. Similarly, the following well-known formulae exist for the Lucas, Jacobsthal, Pell, Pell-Lucas, and Jacobsthal-Lucas numbers (see [5]), respectively:…”

Section: Some Sequences With Property (11)mentioning

confidence: 96%

Notes on a General Sequence

Farhadian

Jakimczuk

2020

Annales Mathematicae Silesianae

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Let {rn}n∈𝕅 be a strictly increasing sequence of nonnegative real numbers satisfying the asymptotic formula rn ~ αβn, where α, β are real numbers with α > 0 and β > 1. In this note we prove some limits that connect this sequence to the number e. We also establish some asymptotic formulae and limits for the counting function of this sequence. All of the results are applied to some well-known sequences in mathematics.

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Section: Some Sequences With Property (11)mentioning

confidence: 96%

Notes on a General Sequence

Farhadian

Jakimczuk

2020

Annales Mathematicae Silesianae

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“…This sequence has been modified in a variety of ways. The Fibonacci Polynomial [5] and the Extended Generalized Fibonacci Polynomial [8] are two such extensions that will be used in this paper. The Fibonacci Polynomial F n (x) is defined by the recurrence relation shown below,…”

Section: Introductionmentioning

confidence: 99%

An Extended Generalized Fibonacci Polynomial Based Coding Method With Error Detection and Correction

Billore¹,

Patel²,

Makwana³

2023

jnanabha

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A Fibonacci coding method is introduced using Extended Generalized Fibonacci Polynomials in this paper. A new square matrix Qⁿm (a, b), the nᵗʰ power of Qm(a, b) of order m × m is defined whose elements are based on extended Generalized Fibonacci Polynomial. Matrix Qⁿ m (a, b) for integer x ≥ 1, a ≥ 1 and b ≥ 1 is considered as the encoding matrix and a matrix Q−ⁿm (a, b) is considered as decoding matrix. An error-detection and error-correction method is also defined in Extended Generalized Fibonacci polynomials.

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“…Bramham and Griffiths in [4] obtained, using combinatorial arguments, a number of convolution identities involving the Jacobsthal and Jacobsthal-Lucas numbers as well as various generalizations of the Fibonacci numbers. Using generating functions, Koshy [10] developed a number of properties for sums of products of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, and Jacobsthal-Lucas numbers. In [15,16], Szakács dealt with convolutions of second order recursive sequences and gave some special convolutions involving the Fibonacci, Pell, Jacobsthal, and Mersenne sequences and their associated sequences.…”

Section: Introductionmentioning

confidence: 99%

Fibonacci–Lucas–Pell–Jacobsthal relations

Frontczak¹,

Goy²,

Shattuck³

2022

AMI

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In this paper, we prove several identities involving linear combinations of convolutions of the generalized Fibonacci and Lucas sequences. Our results apply more generally to broader classes of second-order linearly recurrent sequences with constant coefficients. As a consequence, we obtain as special cases many identities relating exactly four sequences amongst the Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, and Jacobsthal-Lucas number sequences. We make use of algebraic arguments to establish our results, frequently employing the Binet-like formulas and generating functions of the corresponding sequences. Finally, our identities above may be extended so that they include only terms whose subscripts belong to a given arithmetic progression of the non-negative integers.

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Sums of Fibonacci–Pell–Jacobsthal products

Cited by 3 publications

References 3 publications

Notes on a General Sequence

Notes on a General Sequence

An Extended Generalized Fibonacci Polynomial Based Coding Method With Error Detection and Correction

Fibonacci–Lucas–Pell–Jacobsthal relations

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