The notion of an∞-generalized Fibonacci sequence has been introduced in [6], and studied in [1], [7], [9]. This class of sequences defined by linear recurrences of infinite order is an extension of the class of ordinary (weighted) r-generalized Fibonacci sequences (r-GFS, for short) with r finite defined by linear recurrences of r order (for example, see [2], [3], [4], [5], [8] etc.) More preci...

Continuous generalizations of the Fibonacci sequence satisfy ODEs that are formal analogues Friedmann equation describing a spatially homogeneous and isotropic cosmology in general relativity. These analogies presented together with their Lagrangian Hamiltonian formulations an invariant sequence.

The famous problem of determining all perfect powers in the Fibonacci sequence and the Lucas sequence has recently been resolved by three of the present authors. We sketch the proof of this result, and we apply it to show that the only Fibonacci numbers Fn such that Fn ± 1 is a perfect power are 0, 1, 2, 3, 5 and 8. The proof of the Fibonacci Perfect Powers Theorem involves very deep mathematic...

In the present paper, for a positive integer $r$, we study bi-periodic $r$-Fibonacci sequence and its family of companion sequences, $r$-Lucas type $s$ with $1 \leq s r$, which extend classical Fibonacci Lucas sequences. Afterwards, establish link between Furthermore, give their properties as linear recurrence relations, generating functions, explicit formulas Binet forms.

In this paper, we consider the relationships between the sums of the generalized order-k Fibonacci and Lucas numbers and 1-factors of bipartite graphs. 1. Introduction We consider the generalized order k Fibonacci and Lucas numbers. In [1], Er de…ned k sequences of the generalized order k Fibonacci numbers as shown: g n = k X j=1 g n j ; for n > 0 and 1 i k; (1.1) with boundary conditions for 1...

Fibonacci polynomial sequence is an extension of sequence. Here we define a generalizing the integer which enumerates number subsets set [n] including no two consecutive even integers. The associated with polynomials. Some basic properties are obtained.

Although it has been studied extensively, Pascal's triangle remains fascinating to explore and there always seems to be some new aspects that are revealed by looking at it closely. In this paper we shall examine a few nice properties of the so-called Fibonacci diagonals, that is, those slant lines whose entries sum to consecutive terms of the Fibonacci sequence. We adopt throughout our text the...

which evidently has the required property. In [1], a similar example is given for]F109* It is implicit in [1], [12], that such sequences exist in 3Fp if ~Fp contains a so-called Fibonacci Primitive Root, or FPR: see below for definitions. Here we show (Theorem 4.2) that such sequences exist in Wp if and only if Wp contains an FPR; moreover, when Wp does contain an FPR, we show that the only suc...

The Fibonacci sequence F = 0, 1, 1, 2, 3, 5, 8, . . . has intrigued mathematicians for centuries, as it seems there is no end to its many surprising properties. Of particular interest to us are its properties when reduced under a modulus. It is well known, for example, that F (mod m) is periodic, that the zeros are equally spaced, and that each period of F (mod m) contains exactly 1, 2, or 4 ze...