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On the Completeness of the Lucas Sequence
The Fibonacci Quarterly, 1969A sequence of positive integers is said to be complete if every positive integer is the sum of a finite number of distinct terms of the sequence. It is well-known that the Lucas sequence \(\{L_j\}\) where \(L_{n+1}=L_n+L_{n-1}\) for \(n>1\) and \(L_0=2\), \(L_1=1\) is complete. In this paper the author proves that if any term \(L_n\), where \(n>1\), is
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Journal of Number Theory, 2007 Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q) is defined by U_0=0, U_1=1, U_n= P*U_{n-1}-Q*U_{n-2} for n >1. The question of when U_n(P,Q) can be a perfect square has generated interest in the literature. We show that for n=2,...,7, U_n is a square for infinitely many pairs (P,Q) with gcd(P,Q)=1; further, for n=8,...,12, the only non-
N Tzanakis
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Palindromes in Lucas Sequences
Monatshefte f�r Mathematik, 2003Say that \(\{w_n\}\) is a Lucas sequence if \(w_{n+2}= rw_{n+1}+sw_n\) where \(s\neq 0\) and \(r^2+4s\neq 0\). An integer is called a palindrome to base \(b\) if the base \(b\) representation of the integer is left unchanged when the digits are reversed. Let \(P(x)\) denote the number of integers \(n\leq x\) such that \(w_n\) is a base \(b\) palindrome.
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On the intersection of k-Lucas sequences and some binary sequences
Periodica Mathematica Hungarica, 2021Lucas sequence \((L_n)\) is determined by \(L_0=2\); \(L_1=1\); \(L_n=L_{n-1}+L_{n-2}\), if \(n\ge 2\). The paper studies \(k\)-generalized \((k\ge 3)\) Lucas sequence. The sequence starts with the \(k\) terms \(0,\dots,0,2, 1\) and each term afterwards is the sum of the \(k\) preceding terms.
Salah Eddine Rihane, Alain Togbé
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Truncated Lucas Sequence and Its Period
Applied Mathematics and Computation, 2014zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Engin Özkan
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CONGRUENCES CONCERNING LUCAS SEQUENCES
International Journal of Number Theory, 2014Let p be a prime greater than 3. In this paper, by using expansions and congruences for Lucas sequences and the theory of cubic residues and cubic congruences, we establish some congruences for [Formula: see text] and [Formula: see text] modulo p, where [x] is the greatest integer not exceeding x, and m is a rational p-adic integer with m ≢ 0 ( mod p).
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AIP Conference Proceedings, 2014
For positive integers n and k, the k-Lucas sequence is defined by the recurrence relation Ln+1 = kLn+Ln−1 with the initial values L0 = 2, L1 = k. The Lucas sequence and Pell-Lucas sequence are two special cases of the k-Lucas sequence. Using a matrix approach, we uncover some new facts concerning the k-Lucas sequence.
C. K. Ho, Jye-Ying Sia, Chin-Yoon Chong
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For positive integers n and k, the k-Lucas sequence is defined by the recurrence relation Ln+1 = kLn+Ln−1 with the initial values L0 = 2, L1 = k. The Lucas sequence and Pell-Lucas sequence are two special cases of the k-Lucas sequence. Using a matrix approach, we uncover some new facts concerning the k-Lucas sequence.
C. K. Ho, Jye-Ying Sia, Chin-Yoon Chong
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ON GENERALIZED LUCAS AND PELL-LUCAS SEQUENCES
2019In this paper, we define the generaziled Lucas sequences and the Pell-Lucas sequences. Further we give Binet-like formulas, generating function, sums formulas and some important identities which involving the generalized Lucas and Pell-Lucas Numbers.
Tas, Zisan Kusaksiz, TAŞCI, DURSUN
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Mersenne Numbers in Generalized Lucas Sequences
Comptes Rendus De L'Academie Bulgare Des SciencesLet $$k \geq 2$$ be an integer and let $$(L_{n}^{(k)})_{n \geq 2-k}$$ be the $$k$$-generalized Lucas sequence with certain initial $$k$$ terms and each term afterward is the sum of the $$k$$ preceding terms. Mersenne numbers are the numbers of the form $$2^a-1$$, where $$a$$ is any positive integer.
Alaa Altassan
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Notes on the (s, t)-Lucas and Lucas Matrix Sequences.
Ars Comb., 2008In this article, defining the matrix extensions of the Fibonacci and Lucas numbers we start a new approach to derive formulas for some integer numbers which have appeared, often surprisingly, as answers to intricate problems, in conventional and in recreational Mathematics.
Civciv, Hacı, Türkmen, Ramazan
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