scholarly journals A Diophantine equation related to the sum of powers of two consecutive generalized Fibonacci numbers

2015 ◽  
Vol 156 ◽  
pp. 1-14 ◽  
Author(s):  
Ana Paula Chaves ◽  
Diego Marques
Keyword(s):  
Diophantine Equation ◽  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers

scholarly journals A Quadratic Diophantine Equation Involving Generalized Fibonacci Numbers

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1010 ◽  
Author(s):  
Ana Paula Chaves ◽  
Pavel Trojovský
Keyword(s):  
Diophantine Equation ◽  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers ◽  
Quadratic Diophantine Equation ◽  
Positive Integers

The sequence of the k-generalized Fibonacci numbers ( F n ( k ) ) n is defined by the recurrence F n ( k ) = ∑ j = 1 k F n − j ( k ) beginning with the k terms 0 , … , 0 , 1 . In this paper, we shall solve the Diophantine equation F n ( k ) = ( F m ( l ) ) 2 + 1 , in positive integers m , n , k and l.

SOME ALGEBRAICAL, COMBINATORIAL AND ANALYTICAL PROPERTIES OF PARAGRASSMANN VARIABLES

2005 ◽  
Vol 20 (20n21) ◽  
pp. 4797-4819 ◽  
Author(s):  
MATTHIAS SCHORK
Keyword(s):  
Differential Equations ◽  
Fibonacci Numbers ◽  
Delta Function ◽  
Geometrical Interpretation ◽  
Generalized Fibonacci Numbers ◽  
Analytical Properties ◽  
Witten Index ◽  
Grassmann Variables

Some algebraical, combinatorial and analytical aspects of paragrassmann variables are discussed. In particular, the similarity of the combinatorics involved with those of generalized exclusion statistics (Gentile's intermediate statistics) is stressed. It is shown that the dimensions of the algebras of generalized grassmann variables are related to generalized Fibonacci numbers. On the analytical side, some of the simplest differential equations are discussed and a suitably generalized Berezin integral as well as an associated delta-function are considered. Some remarks concerning a geometrical interpretation of recent results about fractional superconformal transformations involving generalized grassmann variables are given. Finally, a quantity related to the Witten index is discussed.

scholarly journals On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of Σn k=0 kW3 k and Σn k=1 kW3− k

2020 ◽  
pp. 37-52 ◽  
Author(s):  
Yuksel Soykan
Keyword(s):  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Second Order ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Special Cases

In this paper, closed forms of the sum formulas Σn k=0 kW3 k and Σn k=1 kW3-k for the cubes of generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Our work generalize second order recurrence relations.

scholarly journals Corrigendum: On Summing Formulas for Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers

2020 ◽  
pp. 66-82
Author(s):  
Y¨uksel Soykan
Keyword(s):  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Summation Formulas ◽  
Special Cases

In this paper, closed forms of the summation formulas for generalized Fibonacci and Gaussian generalized Fibonacci numbers are presented. Then, some previous results are recovered as particular cases of the present results. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers and Gaussian Fibonacci, Gaussian Lucas, Gaussian Pell, Gaussian Pell-Lucas, Gaussian Jacobsthal, Gaussian Jacobsthal-Lucas numbers.

scholarly journals Aitken sequences and generalized Fibonacci numbers

1985 ◽  
Vol 45 (172) ◽  
pp. 553-553 ◽  
Author(s):  
J. H. McCabe ◽  
G. M. Phillips
Keyword(s):  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers
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