Related papers: On certain Fibonacci representations
In this paper, we connect two well established theories, the Fibonacci numbers and the Jordan algebras. We give a series of matrices, from literature, used to obtain recurrence relations of second-order and polynomial sequences. We also…
It is shown that the unique representation of positive integers in terms of tribonacci numbers and the unique representation in terms of iterated A, B and C sequences defined from the tribonacci word are equivalent. Two auxiliary…
We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These…
In this paper, we consider the new family of recurrence sequences of $(q,k)$-generalized Fibonacci numbers. These sequences naturally extend the well-known sequences of $k$-generalized Fibonacci numbers and generalized $k$-order Pell…
We derive several identities for arbitrary homogeneous second order recurrence sequences with constant coefficients. The results are then applied to present a unified study of six well known integer sequences, namely the Fibonacci sequence,…
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…
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.
In this paper, new families of generalized Fibonacci and Lucas numbers are introduced. In addition, we present the recurrence relations and the generating functions of the new families for $k=2$.
We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several…
We study Fibonacci compositions, which are compositions of natural numbers that only use Fibonacci numbers, in two different contexts. We first prove inequalities comparing the number of Fibonacci compositions to regular compositions where…
We study compositions whose parts are colored by subsequences of the Fibonacci numbers. We give explicit bijections between Fibonacci colored compositions and several combinatorial objects, including certain restricted ternary and…
The Fibonacci numbers satisfy the famous recurrence $F_n = F_{n - 1} + F_{n - 2}$. The theory of C-finite sequences ensures that the Fibonacci numbers whose indices are divisible by $m$, namely $F_{mn}$, satisfy a similar recurrence for…
Fibonacci sequence, generated by summing the preceding two terms, is a classical sequence renowned for its elegant properties. In this paper, leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of…
Let $(F_n)_{n\ge 1}$ be the Fibonacci sequence. Define $P(F_n): = (\sum_{i=1}^n F_i)_{n\ge 1}$; that is, the function $P$ gives the sequence of partial sums of $(F_n)$. In this paper, we first give an identity involving $P^k(F_n)$, which is…
In this paper, we define Tribonacci and Tribonacci-Lucas matrix sequences and investigate their properties.
In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving…
We study formal power series which can be interpreted as interpolations of Fibonacci and Lucas polynomials with even (or odd) indices.
A chain is an ordering of the integers 1 to n such that adjacent pairs have sums of a particular form, such as squares, cubes, triangular numbers, pentagonal numbers, or Fibonacci numbers. For example 4 1 2 3 5 form a Fibonacci chain while…
This paper describes a class of sequences that are in many ways similar to Fibonacci sequences: given n, sum the previous two terms and divide them by the largest possible power of n. The behavior of such sequences depends on n. We analyze…
Farey sequences, Stern-Brocot sequences, the Calkin-Wilf sequences are shown to be generated via almost identical second order recurrence relations. These sequences have combinatorial, computational, and geometric applications, and are…