Related papers: Fibonacci numbers and Trivalent graphs

This paper is concerned with developing some new identities of generalized Fibonacci numbers and generalized Pell numbers. A new class of generalized numbers is introduced for this purpose. The two well-known identities of Sury and Marques…

In part 1 of this paper some linear weighted generalized Fibonacci number summation identities were derived using the fact that the Fibonacci number is the residue of a rational function. In this part, using the same method, some quadratic…

This paper introduces a variation on an identity by Bruckman and Good. Using this identity, we are able to derive various well-known sums involving reciprocals of Fibonacci and Lucas numbers, including the case when the indices form an…

Based on the combinatorial interpretation of the ordered Bell numbers, which count all the ordered partitions of the set $[n]=\{1,2,\dots,n\}$, we introduce the Fibonacci partition as a Fibonacci permutation of its blocks. Then we define…

We combinatorially prove Tetranacci, Tetranacci-Fibonacci, and additional identities using only squares and dominoes on a hexagonal double-strip. Some of these are new proofs of old identities, and others we believe have never been seen…

We give an overview about well-known basic properties of two classes of q-Fibonacci and q-Lucas polynomials and offer a common generalization.

In this paper we generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers…

We study matrices which transform the sequence of Fibonacci or Lucas polynomials with even index to those with odd index and vice versa. They turn out to be intimately related to generalized Stirling numbers and to Bernoulli, Genocchi and…

We use probability urn models to discover some known and unknown series identities involving Fibonacci numbers.

The optimal cube factor of a graph, a special kind of component factor, is first introduced. Furthermore, the optimal cube factors of Fibonacci and matchable Lucas cubes are studied; and some results on the Padovan sequence and binomial…

In 2016, Edgar and, independently of him, Bhatnagar sta\-ted a nice polynomial identity that connects Fibonacci and Lucas numbers. Shortly after their publications, this identity has been generalized in two different ways: Dafnis, Phillipou…

Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…

We continue our study on relationships between Fibonacci (Lucas) numbers and Bernoulli numbers and polynomials. The derivations of our results are based on functional equations for the respective generating functions, which in our case are…

Using elementary methods, we establish old and new relations between binomial coefficients, Fibonacci numbers, Lucas numbers, and more.

Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are…

We study formulas expressing Fibonacci numbers as sums over compositions using free submonoids of the free monoid of compositions with parts 1 and 2.

Using a straightforward elementary approach, we derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular…

While there are many identities involving the Euler and Bernoulli numbers, they are usually proved analytically or inductively. We prove two identities involving Euler and Bernoulli numbers with combinatorial reasoning via up-down…

The paper investigates relationship between algebraic expressions and graphs. We consider a digraph called a Fibonacci graph which gives a generic example of non-series-parallel graphs. Our intention in this paper is to simplify the…

We study formal power series which can be interpreted as interpolations of Fibonacci and Lucas polynomials with even (or odd) indices.