Related papers: Compositions and Fibonacci Identities
Utilizing spectral residues of parameterized, recursively defined sequences, we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first…
By using definition of Golden derivative, corresponding Golden exponential function and Fibonomial coefficients, we introduce generating functions for Bernoulli-Fibonacci polynomials and related numbers. Properties of these polynomials and…
Double Fibonacci sequences are introduced and they are related to operations with Fibonacci modules. Generalizations and examples are also discussed.
We study sums of powers of Fibonacci and Lucas polynomials of the form $% \sum_{n=0}^{q}F_{tsn}^{k}(x) $ and $\sum_{n=0}^{q}L_{tsn}^{k}% (x) $, where $s,t,k$ are given natural numbers, together with the corresponding alternating sums…
The aim of this note is to survey the factorizations of the Fibonacci infinite word that make use of the Fibonacci words and other related words, and to show that all these factorizations can be easily derived in sequence starting from…
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 give a combinatorial proof of a formula giving the partial sums of the $k$-bonacci sequence as alternating sums of powers of two multiplied by binomial coefficients. As a corollary we obtain a formula for the $k$-bonacci numbers.
In this paper, we present a new approach to the convolved Fibonacci numbers arising from the generating function of them and give some new and explicit identities for the convolved Fibonacci numbers.
Let $\alpha = (1+\sqrt{5})/2$ and define the lower and upper Wythoff sequences by $a_i = \lfloor i \alpha \rfloor$, $b_i = \lfloor i \alpha^2 \rfloor$ for $i \geq 1$. In a recent interesting paper, Kawsumarng et al. proved a number of…
This paper considers the properties of Tribonacci numbers on identities, matrices, and determinants. In the first front part, we obtain several symmetric identities of Tribonacci numbers by a matrix-based approach and binomial inversion…
We derive various weighted summation identities, including binomial and double binomial identities, for Tribonacci numbers. Our results contain some previously known results as special cases.
We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…
The paper studies algebraic independence of certain reciprocal sums of Fibonacci and Lucas sequences. Also more general binary recurrences are considered. The main tool is Mahler's method reducing the investigation of the algebraic…
Let $f_1=1,f_2=2$ and $f_i=f_{i-1}+f_{i-2}$ for $i>2$ be the sequence of Fibonacci numbers. Let $\Phi_h(n)$ be the quantity of partitions of natural number $n$ into $h$ different Fibonacci numbers. In terms of Zeckendorf partition of $n$ I…
We solve an elementary number theory problem on sums of fractional parts, using methods from group theory. We apply our result to deduce the finiteness of certain monodromy representations.
The focus of this note is to formulate the algorithms and give the examples used by Fibonacci in Liber Abaci to expand any fraction into a sum of unit fractions. The description in Liber Abaci is all verbal and the examples are numbers…
One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer…
We present some new linear, quadratic, cubic and quartic binomial Fibonacci, Lucas and Fibonacci--Lucas summation identities.
In this paper, we analyze the density of the Fibonacci word and its derived forms by examining the morphisms associated with each. It offers a comparative analysis of the density of Fibonacci numbers alongside other words derived from…
We give new identities for some symmetric polynomials. As applications of these identities, we obtain some formulas for a higher order analogue of Fibonacci and Lucas numbers.