Related papers: Distance cube polynomials of Fibonacci and Lucas-r…
In this article, we present a trick around Fibonacci numbers which can be found in several magic books. It consists in computing quickly the sum of the successive terms of a Fibonacci-like sequence. We give explanations and extensions of…
The {\em Fibonacci cube} of dimension $n$, denoted as $\Gamma\_n$, is the subgraph of $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. We study the maximum number of disjoint subgraphs in $\Gamma\_n$ isomorphic to $Q\_k$, and…
In this work, we introduce a symmetric algorithm obtained by the recurrence relation a_{n}^{k}=a_{n-1}^{k}+a_{n}^{k-1}. We point out that this algorithm can be apply to hyperharmonic-, ordinary and incomplete Fibonacci- and Lucas numbers.…
The $d$-Fibonacci digraphs $F(d,k)$, introduced here, have the number of vertices following generalized Fibonacci-like sequences. They can be defined both as digraphs on alphabets and as iterated line digraphs. Here we study some of their…
This survey paper contains a tutorial introduction to distance-regular graphs, with an emphasis on the subconstituent algebra and the $Q$-polynomial property.
In this work, by using Pauli matrices, we introduce four families of polynomials indexed over the positive integers. These polynomials have rational or imaginary rational coefficients. It turns out that two of these families are closely…
The main object of the paper is to reveal connections between Chebyshev polynomials of the first and second kinds and Fibonacci polynomials introduced by Catalan. This is achieved by relating the respective (ordinary and exponential)…
Partial cubes are the graphs which can be embedded into hypercubes. The {\em cube polynomial} of a graph $G$ is a counting polynomial of induced hypercubes of $G$, which is defined as $C(G,x):=\sum_{i\geqslant 0}\alpha_i(G)x^i$, where…
We show that certain weighted Fibonacci and Lucas series can always be expressed as linear combinations of polylogarithms. In some special cases we evaluate the series in terms of Bernoulli polynomials, making use of the connection between…
The Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube $Q_n$ induced by vertices with no consecutive $1$s. Munarini introduced Pell graphs, a variation of Fibonacci cubes defined on ternary strings. A generalization of Pell graphs…
This paper presents new identities expressing the terms of Fibonacci, Lucas, and generalized Fibonacci sequences with multiple indices through powers of Lucas numbers and binomial coefficients. The obtained formulas rely on the application…
In this paper we introduce the polynomials $\{d_n^{(r)}(x)\}$ and $\{D_n^{(r)}(x)\}$ given by $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k} \ (n\ge 0)$, $D_0^{(r)}(x)=1,\ D_1^{(r)}(x)=x$ and…
We explore new types of binomial sums with Fibonacci and Lucas numbers. The binomial coefficients under consideration are $\frac{n}{n+k}\binom{n+k}{n-k}$ and $\frac{k}{n+k}\binom{n+k}{n-k}$. The identities are derived by relating the…
The symmetric numerical semigroups S(F_a,F_b,F_c) and S(L_k,L_m,L_n) generated by three Fibonacci (F_a,F_b,F_c) and Lucas (L_k,L_m,L_n) numbers are considered. Based on divisibility properties of the Fibonacci and Lucas numbers we establish…
This study focuses on the application of deep geometric models to solve the 3x3x3 Rubik's Cube. We begin by discussing the cube's graph representation and defining distance as the model's optimization objective. The distance approximation…
We investigate structural and algorithmic advantages of a directed version of the well-researched class of distance-hereditary graphs. Since the previously defined distance-hereditary digraphs do not permit a recursive structure, we define…
The interlace polynomial q was introduced by Arratia, Bollobas, and Sorkin. It encodes many properties of the orbit of a graph under edge local complementation (ELC). The interlace polynomial Q, introduced by Aigner and van der Holst,…
Using generating functions, we derive many identities involving balancing and Lucas-balancing polynomials. By relating these polynomials to Chebyshev polynomials of the first and second kind, and Fibonacci and Lucas numbers, we offer some…
The $n$-dimensional hypercube graph $Q_n$ has as vertices all subsets of $\{1, \ldots, n\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture states that every matching of the $n$-dimensional…
We present a certain generalization of a recent result of M. I. Cirnu on linear recurrence relations with coefficient in progressions [2]. We provide some interesting examples related to some well-known integer sequences, such as Fibonacci…