Displaying similar documents to “On a generalization of the Pell sequence”

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 .

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 .

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 . ...

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.

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.

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 ) .

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...

Some interpretations of the ( k , p ) -Fibonacci numbers

Natalia Paja, Iwona Włoch (2021)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we consider two parameters generalization of the Fibonacci numbers and Pell numbers, named as the ( k , p ) -Fibonacci numbers. We give some new interpretations of these numbers. Moreover using these interpretations we prove some identities for the ( k , p ) -Fibonacci numbers.

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 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.

On perfect powers in k -generalized Pell sequence

Zafer Şiar, Refik Keskin, Elif Segah Öztaş (2023)

Mathematica Bohemica

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Let k 2 and let ( P n ( k ) ) n 2 - k be the k -generalized Pell sequence defined by P n ( k ) = 2 P n - 1 ( k ) + P n - 2 ( k ) + + P n - k ( k ) for n 2 with initial conditions P - ( k - 2 ) ( k ) = P - ( k - 3 ) ( k ) = = P - 1 ( k ) = P 0 ( k ) = 0 , P 1 ( k ) = 1 . In this study, we handle the equation P n ( k ) = y m in positive integers n , m , y , k such that k , y 2 , and give an upper bound on n . Also, we will show that the equation P n ( k ) = y m with 2 y 1000 has only one solution given by P 7 ( 2 ) = 13 2 .

On the structure of sequences with forbidden zero-sum subsequences

W. D. Gao, R. Thangadurai (2003)

Colloquium Mathematicae

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We study the structure of longest sequences in d which have no zero-sum subsequence of length n (or less). We prove, among other results, that for n = 2 a and d arbitrary, or n = 3 a and d = 3, every sequence of c(n,d)(n-1) elements in d which has no zero-sum subsequence of length n consists of c(n,d) distinct elements each appearing n-1 times, where c ( 2 a , d ) = 2 d and c ( 3 a , 3 ) = 9 .

Padovan and Perrin numbers as products of two generalized Lucas numbers

Kouèssi Norbert Adédji, Japhet Odjoumani, Alain Togbé (2023)

Archivum Mathematicum

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Let P m and E m be the m -th Padovan and Perrin numbers respectively. Let r , s be non-zero integers with r 1 and s { - 1 , 1 } , let { U n } n 0 be the generalized Lucas sequence given by U n + 2 = r U n + 1 + s U n , with U 0 = 0 and U 1 = 1 . In this paper, we give effective bounds for the solutions of the following Diophantine equations P m = U n U k and E m = U n U k , where m , n and k are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing sequences.

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.

Coprimality of integers in Piatetski-Shapiro sequences

Watcharapon Pimsert, Teerapat Srichan, Pinthira Tangsupphathawat (2023)

Czechoslovak Mathematical Journal

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We use the estimation of the number of integers n such that n c belongs to an arithmetic progression to study the coprimality of integers in c = { n c } n , c > 1 , c .

Cobham's theorem for substitutions

Fabien Durand (2011)

Journal of the European Mathematical Society

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The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let α and β be two multiplicatively independent Perron numbers. Then a sequence x A , where A is a finite alphabet, is both α -substitutive and β -substitutive if and only if x is ultimately...

On the distribution of ( k , r ) -integers in Piatetski-Shapiro sequences

Teerapat Srichan (2021)

Czechoslovak Mathematical Journal

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A natural number n is said to be a ( k , r ) -integer if n = a k b , where k > r > 1 and b is not divisible by the r th power of any prime. We study the distribution of such ( k , r ) -integers in the Piatetski-Shapiro sequence { n c } with c > 1 . As a corollary, we also obtain similar results for semi- r -free integers.

On the Diophantine equation j = 1 k j F j p = F n q

Gökhan Soydan, László Németh, László Szalay (2018)

Archivum Mathematicum

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Let F n denote the n t h term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation F 1 p + 2 F 2 p + + k F k p = F n q in the positive integers k and n , where p and q are given positive integers. A complete solution is given if the exponents are included in the set { 1 , 2 } . Based on the specific cases we could solve, and a computer search with p , q , k 100 we conjecture that beside the trivial solutions only F 8 = F 1 + 2 F 2 + 3 F 3 + 4 F 4 , F 4 2 = F 1 + 2 F 2 + 3 F 3 , and F 4 3 = F 1 3 + 2 F 2 3 + 3 F 3 3 satisfy the title equation.

The number of solutions to the generalized Pillai equation ± r a x ± s b y = c .

Reese Scott, Robert Styer (2013)

Journal de Théorie des Nombres de Bordeaux

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We consider N , the number of solutions ( x , y , u , v ) to the equation ( - 1 ) u r a x + ( - 1 ) v s b y = c in nonnegative integers x , y and integers u , v { 0 , 1 } , for given integers a &gt; 1 , b &gt; 1 , c &gt; 0 , r &gt; 0 and s &gt; 0 . When gcd ( r a , s b ) = 1 , we show that N 3 except for a finite number of cases all of which satisfy max ( a , b , r , s , x , y ) &lt; 2 · 10 15 for each solution; when gcd ( a , b ) &gt; 1 , we show that N 3 except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving N = 3 solutions.