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Let p, q, k and ℓ be positive integers. The (p , q , k , ℓ) -Fibonacci sequence (F k , ℓ , p , q) n ≥ 0 is the four-parameter sequence defined by the following recurrence F k , ℓ , p , q (n) = k F k , ℓ , p , q (n - p) + ℓ F k , ℓ , p , q (n - q) , with appropriate initial conditions. In this paper, we study the geometric, algebraic, and analytic aspects of the roots of the characteristic polynomial of this sequence, namely, f (x) = x q - k x q - p - ℓ . [ABSTRACT FROM AUTHOR]

Let (F n) n be the Fibonacci sequence defined by F n + 2 = F n + 1 + F n with F 0 = 0 and F 1 = 1 . In this paper, we prove that for any integer m ≥ 1 there exists a positive constant C m for which lim n → ∞ { (∑ k = n ∞ 1 F m k 2) − 1 − (F m n 2 − F m (n − 1) 2 + (− 1) m n C m) } = 0. Furthermore, we show that C m tends to 2 / 5 as m → ∞ (indeed, we provide quantitative versions of the previous results as well as an explicit form for C m ). This confirms some questions proposed by Lee and Park [J. Inequal. Appl. 2020(1):91 2020]. [ABSTRACT FROM AUTHOR]

Recently, Bednarz introduced a new two-parameter generalization of the Fibonacci sequence, which is called the (k , p) -Fibonacci sequence and denoted by (F k , p (n)) n ≥ 0 . In this paper, we study the geometry of roots of the characteristic polynomial of this sequence. [ABSTRACT FROM AUTHOR]

Many mathematicians have been interested in the study of recursive sequences. Among them, a class of "chaotic" sequences are named "meta-Fibonacci sequences." The main example of meta-Fibonacci sequence was introduced by Hofstadter, and it is called the Q-sequence. Recently, Alkan–Fox–Aybar and the author studied the pattern induced by the connection between the Q-sequence and other known sequences. Here, we continue this program by studying a "Mertens' version" of the Hofstadter sequence, defined (for x > 0) by x ↦ ∑ n ≤ x μ n Q n , where µ(n) is the Möbius function. In particular, as we shall see, this function encodes many interesting properties which relate prime numbers to "meta-sequences". [ABSTRACT FROM AUTHOR]

Let F n be the n th Fibonacci number and L n be the n th Lucas number. The order of appearance z (n) of a natural number n is defined as the smallest natural number k such that n divides F k . For instance, z (F n) = n , for all n > 2. In this paper, among other things, we prove that z (F m − F n) depends on L p , where p is the greatest common divisor of numbers m and n , which fulfill the condition m ≡ n (mod 2). [ABSTRACT FROM AUTHOR]

This paper is devoted to a generalization of some previous results, as we completely solve a linear homogeneous difference equation of the second order with an exponential coefficient. [ABSTRACT FROM AUTHOR]

For k ≥ 2, the k-generalized Fibonacci sequence ( F ) 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 = k = k with t > 1 and m > k+ 1 ≥ 4 is F = 3. [ABSTRACT FROM AUTHOR]

The order of appearance (in the Fibonacci sequence) function z : Z ≥ 1 → Z ≥ 1 is an arithmetic function defined for a positive integer n as z (n) = min { k ≥ 1 : F k ≡ 0 (mod n) } . A topic of great interest is to study the Diophantine properties of this function. In 1992, Sun and Sun showed that Fermat's Last Theorem is related to the solubility of the functional equation z (n) = z (n 2) , where n is a prime number. In addition, in 2014, Luca and Pomerance proved that z (n) = z (n + 1) has infinitely many solutions. In this paper, we provide some results related to these facts. In particular, we prove that lim sup n → ∞ (z (n + 1) − z (n)) / (log n) 2 − ϵ = ∞ , for all ϵ ∈ (0 , 2) . [ABSTRACT FROM AUTHOR]

Let (F n) n ≥ 0 be the Fibonacci sequence. The order of appearance function (in the Fibonacci sequence) z : Z ≥ 1 → Z ≥ 1 is defined as z (n) : = min { k ≥ 1 : F k ≡ 0 (mod n) } . In this paper, among other things, we prove that z (n) is an even number for almost all positive integers n (i.e., the set of such n has natural density equal to 1). [ABSTRACT FROM AUTHOR]

Let (t n (r)) n ≥ 0 be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence t n (r) = t n − 1 (r) + ⋯ + t n − r (r) for n ≥ r , with initial values t 0 (r) = 0 and t i (r) = 1 , for all 1 ≤ i ≤ r . In 2002, Grossman and Luca searched for terms of the sequence (t n (2)) n , which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any ℓ ≥ 1 , there exists an effectively computable constant C = C (ℓ) > 0 (only depending on ℓ), such that, if (m , n , r) is a solution of t m (r) = n ! + (n + 1) ! + ⋯ + (n + ℓ) ! , with r even, then max { m , n , r } < C . As an application, we solve the previous equation for all 1 ≤ ℓ ≤ 5 . [ABSTRACT FROM AUTHOR]

For any integer k ≥ 2 , the sequence of the k-generalized Fibonacci numbers (or k-bonacci numbers) is defined by the k initial values F − (k − 2) (k) = ⋯ = F 0 (k) = 0 and F 1 (k) = 1 and such that each term afterwards is the sum of the k preceding ones. In this paper, we search for repdigits (i.e., a number whose decimal expansion is of the form a a ... a , with a ∈ [ 1 , 9 ] ) in the sequence (F n (k) F n (k + m)) n , for m ∈ [ 1 , 9 ] . This result generalizes a recent work of Bednařík and Trojovská (the case in which (k , m) = (2 , 1) ). Our main tools are the transcendental method (for Diophantine equations) together with the theory of continued fractions (reduction method). [ABSTRACT FROM AUTHOR]

In this paper, we use Baker's theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can be seen as the easiest case of palindromic numbers, which are a "symmetrical" type of numbers) that can be written in the form F n + T n , for some n ≥ 1 , where (F n) n ≥ 0 and (T n) n ≥ 0 are the sequences of Fibonacci and Tribonacci numbers, respectively. [ABSTRACT FROM AUTHOR]

Let k ≥ 1 be an integer and denote (F k , n) n as the k-Fibonacci sequence whose terms satisfy the recurrence relation F k , n = k F k , n − 1 + F k , n − 2 , with initial conditions F k , 0 = 0 and F k , 1 = 1 . In the same way, the k-Lucas sequence (L k , n) n is defined by satisfying the same recursive relation with initial values L k , 0 = 2 and L k , 1 = k . The sequences (F k , n) n ≥ 0 and (L k , n) n ≥ 0 were introduced by Falcon and Plaza, who derived many of their properties. In particular, they proved that F k , n 2 + F k , n + 1 2 = F k , 2 n + 1 and F k , n + 1 2 − F k , n − 1 2 = k F k , 2 n , for all k ≥ 1 and n ≥ 0 . In this paper, we shall prove that if k > 1 and F k , n s + F k , n + 1 s ∈ (F k , m) m ≥ 1 for infinitely many positive integers n, then s = 2 . Similarly, that if F k , n + 1 s − F k , n − 1 s ∈ (k F k , m) m ≥ 1 holds for infinitely many positive integers n, then s = 1 or s = 2 . This generalizes a Marques and Togbé result related to the case k = 1 . Furthermore, we shall solve the Diophantine equations F k , n = L k , m , F k , n = F n , k and L k , n = L n , k . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The order of appearance z : Z > 0 → Z > 0 is an arithmetic function related to the Fibonacci sequence (F n) n . This function is defined as the smallest positive integer solution of the congruence F k ≡ 0 (mod n) . In this paper, we shall provide lower and upper bounds for the functions ∑ n ≤ x z (n) / n , ∑ p ≤ x z (p) and ∑ p r ≤ x z (p r) . [ABSTRACT FROM AUTHOR]

Let F n be the nth Fibonacci number. Order of appearance z (n) of a natural number n is defined as smallest natural number k, such that n divides F k . In 1930, Lehmer proved that all solutions of equation z (n) = n ± 1 are prime numbers. In this paper, we solve equation z (n) = n + ℓ for | ℓ | ∈ { 1 , ... , 9 } . Our method is based on the p-adic valuation of Fibonacci numbers. [ABSTRACT FROM AUTHOR]

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 . [ABSTRACT FROM AUTHOR]