Identities of Generalized Fibonacci-Like Sequence

Singh, Mamta; Sikhwal, Omprakash; Gupta, Yash P.

doi:10.12691/tjant-2-5-3

Cited by 4 publications

(4 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The reader shall see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] for some works on number theory and its applications.…”

Section: Resultsmentioning

confidence: 99%

Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

Mamombe¹

2019

JAMCS

View full text Add to dashboard Cite

The proportiones perfectus law states that σ_x^y=(x+√(x^2+4y))/2 is a proportione perfectus if 1≤y≤x such that for an arbitrary positive integer , there exists an integer sequence defined simultaneously by the quasigeometric relation h_(n+1)=round(σ_x^y h_n ),n≥1 and the arithmetic relation h_(n+2)=〖xh〗_(n+1)+〖yh〗_n,n≥1. When x=y=1 σ_x^y is the golden mean. When x=2,y=1,σ_x^y is the silver mean. In previous works we introduced the theory of number genetics – a framework of logic within which the golden section is studied. In this work we apply the concept to all proportiones perfectus. Let be defined as above. Furthermore, let h1 satisfy {█(round(h_1/(σ_x^y ))=w@round(wσ_x^y)≠h_1 ). ┤Now let H'nbe Hn with h1=1 Again let be defined as A robust universal computing machine is herein developed for the purpose of establishing the relationship h_i=g_(n+i-1)±h_i^',i,n≥1, which relationship is key to the logic protocol. It is clear therefore that this system of logic presents as the building block of every (including itself) within a particular σ_x^y regime. Clearly number genetics takes central place in the proportiones perfectus theory.

show abstract

“…The reader shall see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] for some works on number theory and its applications.…”

Section: Resultsmentioning

confidence: 99%

Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

Mamombe¹

2019

JAMCS

View full text Add to dashboard Cite

show abstract

“…T. Koshy [15] explained two chapters on the use of matrices and determinants. Many determinant identities of generalized Fibonacci sequence are discussed in [4,6] and [11]. In this section some determinant identities of Generalized Fibonacci-Like sequence are presented.…”

Section: Some Determinant Identitiesmentioning

confidence: 99%

Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences

Gupta¹,

Singh²,

Sikhwal³

2016

TJANT

Self Cite

View full text Add to dashboard Cite

show abstract

“…Various generalizations of the aforementioned sequence have been derived since it was first discovered by Fibonacci in the 13 th century. Fibonacci sequence has been generalized mainly by two ways: by maintaining the recurrence relation and varying the initial conditions [1,3,4,5,7,9,10], and by varying the recurrence relation and maintaining the initial conditions [2,4,8,9,11,13,12,15]. Some of the properties that have been obtained by various researchers are not limited to finding a closed form for the n th term of the sequence, sum of the first n terms of the sequence, sum of the first n terms with odd (or even) indices of the sequence, explicit sum formula, Catalan's identity, Cassini's identity, d'Ocagne's identity, Honsberger's identity, determinant identities, and generating function among many others.…”

Section: Introductionmentioning

confidence: 99%

On Generalized Fibonacci Numbers

Oduol

Okoth

2020

Communications in Advanced Mathematical Sciences

View full text Add to dashboard Cite

Fibonacci numbers and their polynomials have been generalized mainly by two ways: by maintaining the recurrence relation and varying the initial conditions, and by varying the recurrence relation and maintaining the initial conditions. In this paper, we introduce and derive various properties of r-sum Fibonacci numbers. The recurrence relation is maintained but initial conditions are varied. Among results obtained are Binet's formula, generating function, explicit sum formula, sum of first n terms, sum of first n terms with even indices, sum of first n terms with odd indices, alternating sum of n terms of r−sum Fibonacci sequence, Honsberger's identity, determinant identities and a generalized identity from which Cassini's identity, Catalan's identity and d'Ocagne's identity follow immediately.

show abstract

Identities of Generalized Fibonacci-Like Sequence

Abstract: The Fibonacci and Lucas sequences are well-known examples of second order recurrence sequences.

Cited by 4 publications

References 11 publications

Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences

On Generalized Fibonacci Numbers

Contact Info

Product

Resources

About