Displaying similar documents to “Some interpretations of the ( k , p ) -Fibonacci numbers”

On the golden number and Fibonacci type sequences

Eugeniusz Barcz (2019)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The paper presents, among others, the golden number ϕ as the limit of the quotient of neighboring terms of the Fibonacci and Fibonacci type sequence by means of a fixed point of a mapping of a certain interval with the help of Edelstein’s theorem. To demonstrate the equality  , where f n is n -th Fibonacci number also the formula from Corollary has been applied. It was obtained using some relationships between Fibonacci and Lucas numbers, which were previously justified.

On a generalization of the Pell sequence

Jhon J. Bravo, Jose L. Herrera, Florian Luca (2021)

Mathematica Bohemica

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The Pell sequence ( P n ) n = 0 is the second order linear recurrence defined by P n = 2 P n - 1 + P n - 2 with initial conditions P 0 = 0 and P 1 = 1 . In this paper, we investigate a generalization of the Pell sequence called the k -generalized Pell sequence which is generated by a recurrence relation of a higher order. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences. Some interesting identities involving the Fibonacci and generalized Pell numbers...

An exponential Diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers

Carlos Alexis Gómez Ruiz, Florian Luca (2014)

Colloquium Mathematicae

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A generalization of the well-known Fibonacci sequence F n 0 given by F₀ = 0, F₁ = 1 and F n + 2 = F n + 1 + F for all n ≥ 0 is the k-generalized Fibonacci sequence F ( k ) n - ( k - 2 ) whose first k terms are 0,..., 0, 1 and each term afterwards is the sum of the preceding k terms. For the Fibonacci sequence the formula F ² + F ² n + 1 ² = F 2 n + 1 holds for all n ≥ 0. In this paper, we show that there is no integer x ≥ 2 such that the sum of the xth powers of two consecutive k-generalized Fibonacci numbers is again a k-generalized Fibonacci number. This...

Some identities involving differences of products of generalized Fibonacci numbers

Curtis Cooper (2015)

Colloquium Mathematicae

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Melham discovered the Fibonacci identity F n + 1 F n + 2 F n + 6 - F ³ n + 3 = ( - 1 ) F . He then considered the generalized sequence Wₙ where W₀ = a, W₁ = b, and W = p W n - 1 + q W n - 2 and a, b, p and q are integers and q ≠ 0. Letting e = pab - qa² - b², he proved the following identity: W n + 1 W n + 2 W n + 6 - W ³ n + 3 = e q n + 1 ( p ³ W n + 2 - q ² W n + 1 ) . There are similar differences of products of Fibonacci numbers, like this one discovered by Fairgrieve and Gould: F F n + 4 F n + 5 - F ³ n + 3 = ( - 1 ) n + 1 F n + 6 . We prove similar identities. For example, a generalization of Fairgrieve and Gould’s identity is W W n + 4 W n + 5 - W ³ n + 3 = e q ( p ³ W n + 4 - q W n + 5 ) .

Fermat k -Fibonacci and k -Lucas numbers

Jhon J. Bravo, Jose L. Herrera (2020)

Mathematica Bohemica

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Using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and Pethő, we find all k -Fibonacci and k -Lucas numbers which are Fermat numbers. Some more general results are given.

Lucas factoriangular numbers

Bir Kafle, Florian Luca, Alain Togbé (2020)

Mathematica Bohemica

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We show that the only Lucas numbers which are factoriangular are 1 and 2 .

A new proof of the q -Dixon identity

Victor J. W. Guo (2018)

Czechoslovak Mathematical Journal

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We give a new and elementary proof of Jackson’s terminating q -analogue of Dixon’s identity by using recurrences and induction.

Pell and Pell-Lucas numbers of the form - 2 a - 3 b + 5 c

Yunyun Qu, Jiwen Zeng (2020)

Czechoslovak Mathematical Journal

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In this paper, we find all Pell and Pell-Lucas numbers written in the form - 2 a - 3 b + 5 c , in nonnegative integers a , b , c , with 0 max { a , b } c .

On the Lucas sequence equations Vₙ = kVₘ and Uₙ = kUₘ

Refik Keskin, Zafer Şiar (2013)

Colloquium Mathematicae

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Let P and Q be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by U₀ = 0, U₁ = 1 and U n + 1 = P U - Q U n - 1 for n ≥ 1, and V₀ = 2, V₁ = P and V n + 1 = P V - Q V n - 1 for n ≥ 1, respectively. In this paper, we assume that P ≥ 1, Q is odd, (P,Q) = 1, Vₘ ≠ 1, and V r 1 . We show that there is no integer x such that V = V r V x ² when m ≥ 1 and r is an even integer. Also we completely solve the equation V = V V r x ² for m ≥ 1 and r ≥ 1 when Q ≡ 7 (mod 8) and x is an even integer. Then we show that when P ≡ 3 (mod 4) and...

On the intersection of two distinct k -generalized Fibonacci sequences

Diego Marques (2012)

Mathematica Bohemica

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Let k 2 and define F ( k ) : = ( F n ( k ) ) n 0 , the k -generalized Fibonacci sequence whose terms satisfy the recurrence relation F n ( k ) = F n - 1 ( k ) + F n - 2 ( k ) + + F n - k ( k ) , with initial conditions 0 , 0 , , 0 , 1 ( k terms) and such that the first nonzero term is F 1 ( k ) = 1 . The sequences F : = F ( 2 ) and T : = F ( 3 ) are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation F n ( k ) = F m ( ) . In this note, we use transcendental tools to provide a general method for finding the intersections F ( k ) F ( m ) which gives...

Repdigits in generalized Pell sequences

Jhon J. Bravo, Jose L. Herrera (2020)

Archivum Mathematicum

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For an integer k 2 , let ( n ) n be the k - generalized Pell sequence which starts with 0 , ... , 0 , 1 ( k terms) and each term afterwards is given by the linear recurrence n = 2 n - 1 + n - 2 + + n - k . In this paper, we find all k -generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence ( P n ( 2 ) ) n . ...

On the least common multiple of Lucas subsequences

Shigeki Akiyama, Florian Luca (2013)

Acta Arithmetica

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We compare the growth of the least common multiple of the numbers u a 1 , . . . , u a n and | u a 1 u a n | , where ( u n ) n 0 is a Lucas sequence and ( a n ) n 0 is some sequence of positive integers.

Diophantine approximations with Fibonacci numbers

Victoria Zhuravleva (2013)

Journal de Théorie des Nombres de Bordeaux

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Let F n be the n -th Fibonacci number. Put ϕ = 1 + 5 2 . We prove that the following inequalities hold for any real α : 1) inf n | | F n α | | ϕ - 1 ϕ + 2 , 2) lim inf n | | F n α | | 1 5 , 3) lim inf n | | ϕ n α | | 1 5 . These results are the best possible.

Bartz-Marlewski equation with generalized Lucas components

Hayder R. Hashim (2022)

Archivum Mathematicum

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Let { U n } = { U n ( P , Q ) } and { V n } = { V n ( P , Q ) } be the Lucas sequences of the first and second kind respectively at the parameters P 1 and Q { - 1 , 1 } . In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation x 2 - 3 x y + y 2 + x = 0 , where ( x , y ) = ( U i , U j ) or ( V i , V j ) with i , j 1 . Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.

Towards Bauer's theorem for linear recurrence sequences

Mariusz Skałba (2003)

Colloquium Mathematicae

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Consider a recurrence sequence ( x k ) k of integers satisfying x k + n = a n - 1 x k + n - 1 + . . . + a x k + 1 + a x k , where a , a , . . . , a n - 1 are fixed and a₀ ∈ -1,1. Assume that x k > 0 for all sufficiently large k. If there exists k₀∈ ℤ such that x k < 0 then for each negative integer -D there exist infinitely many rational primes q such that q | x k for some k ∈ ℕ and (-D/q) = -1.

Lucas sequences and repdigits

Hayder Raheem Hashim, Szabolcs Tengely (2022)

Mathematica Bohemica

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Let ( G n ) n 1 be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are { U n } and { V n } , respectively. We show that the Diophantine equation G n = B · ( g l m - 1 ) / ( g l - 1 ) has only finitely many solutions in n , m + , where g 2 , l is even and 1 B g l - 1 . Furthermore, these solutions can be effectively determined by reducing such equation to biquadratic elliptic curves. Then, by a result of Baker (and its best improvement due to Hajdu and Herendi) related to the bounds of the integral...