This paper concerns the properties of generalized bi-periodic Fibonacci numbers {Gn} generated from recurrence relation: Gn=aGn−1+Gn−2 (n is even) or Gn=bGn−1+Gn−2 odd). We derive general identities for reciprocal sums products two numbers. More precisely, we obtain formulas integer parts ∑k=n∞(a/b)ξ(k+1)GkGk+m−1,m=0,2,4,⋯, and ∑k=n∞1GkGk+m−1,m=1,3,5,⋯.

The/?-adic order, vp(r), of r is the exponent of the highest power of a prime/? which divides r. We characterize the/?-adic order vp(Fn) of the F„ sequence using multisection identities. The method of multisection is a helpful tool in discovering and proving divisibility properties. Here it leads to invariants of the modulo p Fibonacci generating function for p ^ 5. The proof relies on some sim...

This paper is concerned with the generalized Euler polynomial matrix $\E^{(\alpha)}(x)$ and $\E$. Taking into account some properties of polynomials numbers, we deduce product formulae for determine inverse We establish explicit expressions $\E(x)$, which involving Pascal, Fibonacci Lucas matrices, respectively. From these get new interesting identities numbers. Also, provide factorizations in ...

An overview is given of effective gravitational field theories with fixed background fields that break spacetime symmetry. The behavior of the background fields and the types of excitations that can occur depend on whether the symmetry breaking is explicit or spontaneous. For example, when the breaking is spontaneous, the background field is dynamical and massless Nambu–Goldstone and massive Hi...

In this paper, we introduce and investigate the generalized Olivier sequences deal with, in detail, two special cases, namely, Olivier-Lucas sequences. We present Binet's formulas, generating functions, Simson summation formulas for these also provide variousmatrices identities associated with Furthermore, show that there are close relations between Olivier, adjusted Pell-Padovan, third order L...

The Fibonacci (p, r)-cube is an interconnection topology, which includes a wide range of connection topologies as its special cases, such as the Fibonacci cube, the postal network, etc. Klavžar and Žigert [S. Klavžar, P. Žigert, Fibonacci cubes are the resonance graphs of fibonaccenes, Fibonacci Quart. 43 (2005) 269–276] proved that Fibonacci cubes are just the Z-transformation graphs (also cal...

The aim of this paper is to construct general forms ordinary generating functions for special numbers and polynomials involving Fibonacci type polynomials, Lucas Chebyshev Sextet Humbert-type chain anti-chain rank the lattices, length any alphabet words, partitions, other graph polynomials. By applying Euler transform Lambert series these functions, many new identities relations are derived. us...

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We first generalize an identity involving the generalized Fibonacci numbers and then apply it to establish some general identities concerning special sums. We also give a sufficient condition on a generalized Fibonacci sequence {Un} such that Un is divisible by an arbitrary prime r for some 2 < n ≤ r − 2. 1. Preliminaries The generalized Fibonacci and Lucas numbers are defined, respectively, by...

This paper presents both a result and method. The closed formula for the sum of first $$m+1,m \ge 0,$$ squares sequence $$F^{(k)}$$ where each member is previous k members with initial conditions $$k-1$$ zeroes followed by 1. generalized includes known sums Fibonacci numbers recent results Ohtsuka-Jakubczyk, Howard-Cooper, Schumacher, Prodinger-Selkirk cases $$k=2,3,4,5,6.$$ contributes coeffic...