scholarly journals A note on Pell-Padovan numbers and their connection with Fibonacci numbers

2020 ◽  
Vol 12 (2) ◽  
pp. 280-288
Author(s):  
T.P. Goy ◽  
S.V. Sharyn
Keyword(s):  
Fibonacci Numbers ◽  
Hessenberg Matrices ◽  
Sums Of Products ◽  
Fibonacci Sequences ◽  
Multinomial Coefficients ◽  
Connection Formulas

In this paper, we find new relations involving the Pell-Padovan sequence which arise as determinants of certain families of Toeplitz-Hessenberg matrices. These determinant formulas may be rewritten as identities involving sums of products of Pell-Padovan numbers and multinomial coefficients. In particular, we establish four connection formulas between the Pell-Padovan and the Fibonacci sequences via Toeplitz-Hessenberg determinants.

scholarly journals Some Properties of Convolved k-Fibonacci Numbers

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
José L. Ramírez
Keyword(s):  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Hessenberg Matrices ◽  
Combinatorial Formulas

We define the convolved k-Fibonacci numbers as an extension of the classical convolved Fibonacci numbers. Then we give some combinatorial formulas involving the k-Fibonacci and k-Lucas numbers. Moreover we obtain the convolved k-Fibonacci numbers from a family of Hessenberg matrices.

On sums related to the numerator of generating functions for the kth power of Fibonacci numbers on sums related to the numerator of generating functions for the kth power of Fibonacci numbers

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Pavel Pražák ◽  
Pavel Trojovský
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Similar Type ◽  
Lucas Numbers ◽  
Fibonomial Coefficients ◽  
Sums Of Products

AbstractNew results about some sums s n(k, l) of products of the Lucas numbers, which are of similar type as the sums in [SEIBERT, J.—TROJOVSK Ý, P.: On multiple sums of products of Lucas numbers, J. Integer Seq. 10 (2007), Article 07.4.5], and sums σ(k) = $$ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} $$ are derived. These sums are related to the numerator of generating function for the kth powers of the Fibonacci numbers. s n(k, l) and σ(k) are expressed as the sum of the binomial and the Fibonomial coefficients. Proofs of these formulas are based on a special inverse formulas.

scholarly journals Extended Fibonacci numbers and polynomials with probability applications

2004 ◽  
Vol 2004 (50) ◽  
pp. 2681-2693
Author(s):  
Demetrios L. Antzoulakos
Keyword(s):  
Generating Function ◽  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Multinomial Coefficients

The extended Fibonacci sequence of numbers and polynomials is introduced and studied. The generating function, recurrence relations, an expansion in terms of multinomial coefficients, and several properties of the extended Fibonacci numbers and polynomials are obtained. Interesting relations between them and probability problems which take into account lengths of success and failure runs are also established.

scholarly journals Psychoacoustic Properties of Fibonacci Sequences

10.14311/1027 ◽  
2008 ◽  
Vol 48 (4) ◽  
Author(s):  
J. Sokoll ◽  
S. Fingerhuth
Keyword(s):  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Binary Sequence ◽  
Fibonacci Sequence ◽  
Almost Periodic ◽  
Listening Tests ◽  
Spectral Components ◽  
Self Similar ◽  
Set Up ◽  
Fibonacci Sequences

1202, Fibonacci set up one of the most interesting sequences in number theory. This sequence can be represented by so-called Fibonacci Numbers, and by a binary sequence of zeros and ones. If such a binary Fibonacci Sequence is played back as an audio file, a very dissonant sound results. This is caused by the “almost-periodic”, “self-similar” property of the binary sequence. The ratio of zeros and ones converges to the golden ratio, as do the primary and secondary spectral components intheir frequencies and amplitudes. These Fibonacci Sequences will be characterized using listening tests and psychoacoustic analyses. 

Representation of integers by k-generalized Fibonacci sequences and applications in cryptography

2021 ◽  
pp. 2150157
Author(s):  
Salim Badidja ◽  
Ahmed Ait Mokhtar ◽  
Özen Özer
Keyword(s):  
Positive Integer ◽  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers ◽  
Representation Of Integers ◽  
Tribonacci Numbers ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences ◽  
Positive Integers

The aim of this paper is to construct a relation between tribonacci numbers and generalized tribonacci numbers. Besides, certain conditions are obtained to generalize the representation of a positive integer [Formula: see text] which is determined in [S. Badidja and A. Boudaoud, Representation of positive integers as a sum of distinct tribonacci numbers, J. Math. Statistic. 13 (2017) 57–61] for a [Formula: see text]-generalized Fibonacci numbers [Formula: see text]. Lastly, some applications to cryptography are given by using [Formula: see text].

scholarly journals On the Reciprocal Sums of Products of Two Generalized Bi-Periodic Fibonacci Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 178
Author(s):  
Younseok Choo
Keyword(s):  
Recurrence Relation ◽  
Fibonacci Numbers ◽  
Sums Of Products ◽  
Reciprocal Sums

This paper concerns the properties of the generalized bi-periodic Fibonacci numbers {Gn} generated from the recurrence relation: Gn=aGn−1+Gn−2 (n is even) or Gn=bGn−1+Gn−2 (n is odd). We derive general identities for the reciprocal sums of products of two generalized bi-periodic Fibonacci numbers. More precisely, we obtain formulas for the integer parts of the numbers ∑k=n∞(a/b)ξ(k+1)GkGk+m−1,m=0,2,4,⋯, and ∑k=n∞1GkGk+m−1,m=1,3,5,⋯.

Generalization of the Fibonacci Sequence to n Dimensions

1966 ◽  
Vol 18 ◽  
pp. 332-349 ◽  
Author(s):  
George N. Raney
Keyword(s):  
Free Semigroup ◽  
Jacobi Polynomials ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Natural Generalization ◽  
Unimodular Group ◽  
Natural Extensions ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences

We introduce certain n X n matrices with integral elements that constitute a free semigroup with identity and generate the n-dimensional unimodular group. In terms of these matrices we define a certain sequence of n-dimensional vectors, which we show is the natural generalization to n dimensions of the Fibonacci sequence. Connections between the generalized Fibonacci sequences and certain Jacobi polynomials are found. The various basic identities concerning the Fibonacci numbers are shown to have natural extensions to n dimensions, and in some cases the proofs are rendered quite brief by the use of known theorems on matrices.

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