The Binet formula, sums and representations of generalized Fibonacci p-numbers
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The generalized Fibonacci p-numbers are considered and the generalized Binet formula, sums, combinatorial representations and generating function are given and an explicit formula for the sums is derived.Abstract:
In this paper, we consider the generalized Fibonacci p-numbers and then we give the generalized Binet formula, sums, combinatorial representations and generating function of the generalized Fibonacci p-numbers. Also, using matrix methods, we derive an explicit formula for the sums of the generalized Fibonacci p-numbers.read more
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Journal ArticleDOI
A new family of k-Fibonacci numbers
TL;DR: The recurrence relations and the generating functions of the new family for k=2 and k=3 are described, and a few identity formulas for the family and the ordinary Fibonacci numbers are presented.
Journal ArticleDOI
Powers of two as sums of two k-Fibonacci numbers
TL;DR: In this article, the Baker-Davenport reduction method in diophantine approximation was used to search for powers of 2 which are sums of two $k-$Fibonacci numbers.
Journal ArticleDOI
On k-Fibonacci numbers of arithmetic indexes
Sergio Falcon,Ángel Plaza +1 more
TL;DR: The sums of k-Fibonacci numbers with indexes in an arithmetic sequence, say an þ r for fixed integers a and r, are studied to give in a straightforward way several formulas for the sums of such numbers.
Journal ArticleDOI
Inversion of k-tridiagonal matrices with Toeplitz structure
Ji-Teng Jia,Ji-Teng Jia,Ji-Teng Jia,Tomohiro Sogabe,Tomohiro Sogabe,Tomohiro Sogabe,Moawwad El-Mikkawy,Moawwad El-Mikkawy,Moawwad El-Mikkawy +8 more
TL;DR: A theoretical result is obtained that under certain assumptions the explicit inverse of a k-tridiagonal Toeplitz matrix can be derived immediately.
Journal ArticleDOI
k-Fibonacci sequences modulo m
Sergio Falcon,Ángel Plaza +1 more
TL;DR: In this paper, the Pisano period of the k-Fibonacci cyclic sequences is studied and the period length is shown to be π k 2 + 4 for every odd number k.
References
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Book
Fibonacci and Lucas Numbers
TL;DR: Hoggatt as discussed by the authors presented a 92-page booklet in its Enrichment Series entitled "Fibonacci and Lucas Numbers" by V. E. Hoggatt.
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The k-Fibonacci sequence and the Pascal 2-triangle
Sergio Falcon,Ángel Plaza +1 more
TL;DR: In this paper, the general k-Fibonacci sequence was found by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge partition.
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Applications of Fibonacci Numbers.
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Connections: The Geometric Bridge Between Art and Science
TL;DR: In this article, the golden mean graphs tilings with polygons two-dimensional networks and lattices polyhedra were compared. And the similarity of the two types of tilings was measured in terms of the proportion in architecture similarity.