K-generalized Fibonacci Sequence Research Articles

The Complete Solution of the Diophantine Equation Fn+1(k)x-Fn-1(k)x=Fm(k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( F_{n+1}^{(k)}\\right) ^x - \\left( F_{n-1}^{(k)}\\right) ^x = F_{m}^{(k)}$$\\end{document}

The well-known Fibonacci sequence has several generalizations, among them, the k-generalized Fibonacci sequence denoted by F(k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F^{(k)}$$\\end{document}. The first k terms of this generalization are 0,…,0,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0, \\ldots , 0, 1$$\\end{document} and each one afterward corresponds to the sum of the preceding k terms. For the Fibonacci sequence the formula Fn+12-Fn-12=F2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{n+1}^2 - F_{n-1}^2 = F_{2n}$$\\end{document} holds for every n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 1$$\\end{document}. In this paper, we study the above identity on the k-generalized Fibonacci sequence terms, completing the work done by Bensella et al. (On the exponential Diophantine equation (Fm+1(k))x-(Fm-1(k))x=Fn(k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(F_{m+1}^{(k)})^x - (F_{m-1}^{(k)})^x = F_n^{(k)}$$\\end{document}, 2022. arxiv:2205.13168).

Upper bound on the solution to F(2k)n = +F(2k)m with negative subscripts

In this paper, we provide an explicit upper bound on the absolute value of the solutions n < m < 0 to the Diophantine equation F(k)n = ±F(k)m, assuming k is even. Here {F(k)n}n ∈ Z denotes the k-generalized Fibonacci sequence. The upper bound depends only on k.

Almost Repdigit k-Fibonacci Numbers with an Application of k-Generalized Fibonacci Sequences

In this paper, we define the notion of almost repdigit as a positive integer whose digits are all equal except for at most one digit, and we search all terms of the k-generalized Fibonacci sequence which are almost repdigits. In particular, we find all k-generalized Fibonacci numbers which are powers of 10 as a special case of almost repdigits. In the second part of the paper, by using the roots of the characteristic polynomial of the k-generalized Fibonacci sequence, we introduce k-generalized tiny golden angles and show the feasibility of this new type of angles in application to magnetic resonance imaging.

Generalized Fibonacci numbers and extreme value laws for the Rényi map

In this paper we prove an extreme value law for a stochastic process obtained by iterating the Rényi map x↦βx(mod1), where we assume that β>1 is an integer. Haiman (2018) derived a recursion formula for the Lebesgue measure of threshold exceedance sets. We show how this recursion formula is related to a rescaled version of the k-generalized Fibonacci sequence. For the latter sequence we derive a Binet formula which leads to a closed-form expression for the distribution of partial maxima of the stochastic process. The proof of the extreme value law is completed by deriving sharp bounds for the dominant root of the characteristic polynomial associated with the Fibonacci sequence.

On the zero-multiplicity of the k-generalized Fibonacci sequence

The k-generalized Fibonacci sequence is the linear recurrent sequence of order k, whose first k terms are and each term afterwards is the sum of the preceding k terms. In this paper, we extend to negative indices and give an upper bound on the absolute value of its largest zero in terms of k. In particular, we find that the zero-multiplicity of the k-generalized Fibonacci sequence is exactly for all . We prove that the zero multiplicity of the k-generalized Fibonacci sequence is at least for all .

K−Fibonacci numbers close to a power of 2

A generalization of the well-known Fibonacci sequence is the k-generalized Fibonacci sequence whose first k terms are 0, . , 0, 1 and each term afterwards is the sum of the preceding k terms. In this paper, by using a lower bound to linear forms in logarithms of algebraic numbers due to Matveev and some argument of the theory of continued fractions, we find all the members of F (k) which are close to a power of 2. This paper continues and extends the previous work of Chern and Cui which investigated the Fibonacci numbers close to a power of 2.

On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes

The k-generalized Fibonacci sequence ( F n ( k ) ) n (sometimes also called k-bonacci or k-step Fibonacci sequence), with k ≥ 2 , is defined by the values 0 , 0 , … , 0 , 1 of starting k its terms and such way that each term afterwards is the sum of the k preceding terms. This paper is devoted to the proof of the fact that the Diophantine equation F m ( k ) = m t , with t > 1 and m > k + 1 , has only solutions F 12 ( 2 ) = 12 2 and F 9 ( 3 ) = 9 2 .

Multiplicative Independence in k-Generalized Fibonacci Sequences

A generalization of the Fibonacci sequence is the k-generalized Fibonacci sequence (F () )n ≥ 2 − k with some fixed integer k ≥ 2 whose first k terms are 0,…, 0, 1 and each term afterward is the sum of the preceding k terms. Carmichael’s primitive divisor theorem ensures that all members after the twelfth of the Fibonacci sequence are multiplicatively independent. Although there is no version of this theorem for k-generalized Fibonacci sequences with k > 2, here we find all the pairs of k-Fibonacci numbers that are multiplicatively dependent.

Terms of Generalized Fibonacci Sequences That are Powers of Their Orders

For k ≥ 2, the k-generalized Fibonacci sequence (F n (k) ) n is defined by the initial values 0,0,…, 0, 1 (k terms) so that each term afterward is the sum of the k preceding terms. In this paper, we prove that the only solution of the Diophantine equation F m (k) = k t = k t with t > 1 and m > k+ 1 ≥ 4 is F 9 (3) = 34.

On some combinations of k-nacci numbers

For k ≥ 2, the k-generalized Fibonacci sequence (Fn(k))n is defined by the initial values 0,0,…,0,1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In this paper, we shall study for which x, k and t the expression xtFn+t(k)+⋯+xFn+1(k)+Fn(k) belongs to (Fm(k))m for infinitely many integers n. This work generalizes [13, Theorem 2] which is related to the case t=1.

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

Let (Fn)n≥0 be the Fibonacci sequence given by Fm+2=Fm+1+Fm, for m≥0, where F0=0 and F1=1. In 2011, Luca and Oyono proved that if Fms+Fm+1s is a Fibonacci number, with m≥2, then s=1 or 2. A well-known generalization of the Fibonacci sequence, is the k-generalized Fibonacci sequence (Fn(k))n which is defined by the initial values 0,0,…,0,1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In this paper, we generalize Luca and Oyono's method by proving that the Diophantine equation(Fm(k))s+(Fm+1(k))s=Fn(k) has no solution in positive integers n,m,k and s, if 3≤k≤min⁡{m,log⁡s}.

On the largest prime factor of the ratio of two generalized Fibonacci numbers

A generalization of the well-known Fibonacci sequence is the k-generalized Fibonacci sequence (Fn(k))n≥2−k for some integer k≥2, whose first k terms are 0,…,0,1 and each term afterwards is the sum of the preceding k terms. In this paper, we look at the prime factors of the reduced rational number Fn(k)/Fm(ℓ) as max⁡{m,n,k,ℓ} tends to infinity.

Generalized Fibonacci numbers of the form 2 a + 3 b + 5 c

For k ≥ 2, the k-generalized Fibonacci sequence (Fn(k))n is defined by the initial values 0, 0,..., 0, 1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In this paper, we find all generalized Fibonacci numbers written in the form 2a + 3b + 5c. This work generalizes a recent Marques-Togbe result [20] concerning the case k = 2.

The proof of a conjecture concerning the intersection of k-generalized Fibonacci sequences

For k ≥ 2, the k-generalized Fibonacci sequence (F ) n is defined by the initial values 0, 0, …, 0,1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In 2005, Noe and Post conjectured that the only solutions of Diophantine equation F = F (l) , with l > k > 1, n > l + 1, m > k + 1 are $(m,n,\ell ,k) = (7,6,3,2)and(12,11,7,3)$ . In this paper, we confirm this conjecture.

ON THE LARGEST PRIME FACTOR OF THE k-FIBONACCI NUMBERS

Let P(m) denote the largest prime factor of an integer m ≥ 2, and put P(0) = P(1) = 1. For an integer k ≥ 2, let [Formula: see text] be the k-generalized Fibonacci sequence which starts with 0, …, 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms. Here, we show that if n ≥ k+2, then [Formula: see text]. Furthermore, we determine all the k-Fibonacci numbers [Formula: see text] whose largest prime factor is less than or equal to 7.

On the k-generalized Fibonacci numbers and high-order linear recurrence relations

In this paper, using elementary symmetric function and complete homogeneous symmetric function, we obtain a determinant formula for the k-generalized Fibonacci sequence. The relationship between the k-generalized Fibonacci sequence and the order- k linear homogeneous recurrence relation has been investigated, and a general solution for the latter is also derived. Furthermore, we obtain an explicit expression for the elements in the nth power of the companion matrix in terms of k-generalized Fibonacci numbers.

On the K-generalized Fibonacci matrix Qk

The k-generalized Fibonacci sequence { g ( k) n } is defined as follows: g ( k) 1 = … = g ( k) k − 2 = 0, g ( k) k − 1 = g ( k) k = 1, and for n > k ⩾ 2, g ( k) n = g ( k) n − 1 + g ( k) n − 2 + … g ( k) n − k . We consider the relationship between g ( k) n and 1-factors of a bipartite graph and the eigenvalues of k-generalized Fibonacci matrix Q k for k ⩾ 2. We give some interesting examples in combinatorics and probability with respect to the k-generalized Fibonacci sequence.