In this data set we first 1000 k Fibonacci numbers, this approach is totally new and never tried ... more In this data set we first 1000 k Fibonacci numbers, this approach is totally new and never tried in k− Fibonacci number literature.
In this paper, we de¯ned new relationship between k Lucas sequences and determinants of their ass... more In this paper, we de¯ned new relationship between k Lucas sequences and determinants of their associated matrices, this approach is di®erent and never tried in k Fibonacci sequence literature.
Coupled Fibonacci sequences involve two sequences of integers in which the elements of one sequen... more Coupled Fibonacci sequences involve two sequences of integers in which the elements of one sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first introduced coupled Fibonacci sequences of second order in additive form. In this paper, I present some properties of multiplicative coupled Fibonacci sequences of fourth order under two specific schemes.
The Fibonacci polynomial has been generalized in many ways,some by preserving the initial conditi... more The Fibonacci polynomial has been generalized in many ways,some by preserving the initial conditions,and others by preserving the recurrence relation.In this article,we study new generalization {M n }(x), with initial conditions M 0 (x) = 2 and M 1 (x) = m(x) + k(x), which is generated by the recurrence relation M n+1 (x) = k(x)M n (x) + M n−1 (x) for n ≥ 2, where k(x), m(x) are polynomials with real coefficients.We produce an extended Binet's formula for {M n }(x) and,thereby identities such as Simpson's,Catalan's,d'Ocagene's,etc.using matrix algebra.Moreover, we present sum formulas concerning this new generalization.
Coupled Fibonacci sequences of lower order have been generalized in number of ways.In this paper ... more Coupled Fibonacci sequences of lower order have been generalized in number of ways.In this paper the Multiplicative Coupled Fibonacci Sequence has been generalized for r th order with some new interesting properties.
ABSTRACT Coupled Fibonacci sequences involve two sequences of integers in which the elements of o... more ABSTRACT Coupled Fibonacci sequences involve two sequences of integers in which the elements of one sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first introduced coupled Fibonacci sequences of second order in additive form. In this paper, I present some properties of multiplicative coupled Fibonacci sequences of fourth order under two specific schemes.
The main purpose of this paper is to study some sums of powers of Generalized k− Fibonacci like s... more The main purpose of this paper is to study some sums of powers of Generalized k− Fibonacci like sequence {M k,n },with initial conditions M k,0 = 2 and M k,1 = m + k,and give several interesting identities.
The Fibonacci sequence has been generalized in many ways,some by preserving the initial condition... more The Fibonacci sequence has been generalized in many ways,some by preserving the initial conditions,and others by preserving the recurrence relation.In this article,we study new generalization {M k,n },with initial conditionsM k,0 = 2 and M k,1 = m + k,which is generated by the recurrence relation M k,n+1 = kM k,n + M k,n−1 for n ≥ 2,where k, m are integer number.We produce sum formulas concerning this new generalization.
In this data set we first 1000 k Fibonacci numbers, this approach is totally new and never tried ... more In this data set we first 1000 k Fibonacci numbers, this approach is totally new and never tried in k− Fibonacci number literature.
In this paper, we de¯ned new relationship between k Lucas sequences and determinants of their ass... more In this paper, we de¯ned new relationship between k Lucas sequences and determinants of their associated matrices, this approach is di®erent and never tried in k Fibonacci sequence literature.
Coupled Fibonacci sequences involve two sequences of integers in which the elements of one sequen... more Coupled Fibonacci sequences involve two sequences of integers in which the elements of one sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first introduced coupled Fibonacci sequences of second order in additive form. In this paper, I present some properties of multiplicative coupled Fibonacci sequences of fourth order under two specific schemes.
The Fibonacci polynomial has been generalized in many ways,some by preserving the initial conditi... more The Fibonacci polynomial has been generalized in many ways,some by preserving the initial conditions,and others by preserving the recurrence relation.In this article,we study new generalization {M n }(x), with initial conditions M 0 (x) = 2 and M 1 (x) = m(x) + k(x), which is generated by the recurrence relation M n+1 (x) = k(x)M n (x) + M n−1 (x) for n ≥ 2, where k(x), m(x) are polynomials with real coefficients.We produce an extended Binet's formula for {M n }(x) and,thereby identities such as Simpson's,Catalan's,d'Ocagene's,etc.using matrix algebra.Moreover, we present sum formulas concerning this new generalization.
Coupled Fibonacci sequences of lower order have been generalized in number of ways.In this paper ... more Coupled Fibonacci sequences of lower order have been generalized in number of ways.In this paper the Multiplicative Coupled Fibonacci Sequence has been generalized for r th order with some new interesting properties.
ABSTRACT Coupled Fibonacci sequences involve two sequences of integers in which the elements of o... more ABSTRACT Coupled Fibonacci sequences involve two sequences of integers in which the elements of one sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first introduced coupled Fibonacci sequences of second order in additive form. In this paper, I present some properties of multiplicative coupled Fibonacci sequences of fourth order under two specific schemes.
The main purpose of this paper is to study some sums of powers of Generalized k− Fibonacci like s... more The main purpose of this paper is to study some sums of powers of Generalized k− Fibonacci like sequence {M k,n },with initial conditions M k,0 = 2 and M k,1 = m + k,and give several interesting identities.
The Fibonacci sequence has been generalized in many ways,some by preserving the initial condition... more The Fibonacci sequence has been generalized in many ways,some by preserving the initial conditions,and others by preserving the recurrence relation.In this article,we study new generalization {M k,n },with initial conditionsM k,0 = 2 and M k,1 = m + k,which is generated by the recurrence relation M k,n+1 = kM k,n + M k,n−1 for n ≥ 2,where k, m are integer number.We produce sum formulas concerning this new generalization.
Shapes defined by the golden ratio have long been considered aesthetically pleasing in western cu... more Shapes defined by the golden ratio have long been considered aesthetically pleasing in western cultures, reflecting nature's balance between symmetry and asymmetry. The ratio is still used frequently in art and design. The golden ratio is also known as the golden mean, golden section, golden number or divine proportion.It is usually denoted by the Greek letter (phi).
In this data set we first 1000 k Fibonacci numbers, this approach is totally new and never tried ... more In this data set we first 1000 k Fibonacci numbers, this approach is totally new and never tried in k− Fibonacci number literature.
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Papers by Ashok D Godase