Related papers: Tetranacci Identities With Squares, Dominoes, And …
We present a different combinatorial interpretations of Lucas and Gibonacci numbers. Using these interpretations we prove several new identities, and simplify the proofs of several known identities. Some open problems are discussed towards…
We prove some identities for the squares of generalized Tribonacci numbers. Various summation identities involving these numbers are derived.
We extend Fibonacci numbers with arbitrary weights and generalize a dozen Fibonacci identities. As a special case, we propose an elliptic extension which extends the $q$-Fibonacci polynomials appearing in Schur's work. The proofs of most 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…
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…
The number of ways to tile an $n$-board (an $n\times1$ rectangular board) with $(\frac12,\frac12;1)$-, $(\frac12,\frac12;2)$-, and $(\frac12,\frac12;3)$-combs is $T_{n+2}^2$ where $T_n$ is the $n$th tribonacci number. A…
We give convolution identities without binomial coefficients for Tetranacci numbers and convolution identities with binomial coefficients for Tetranacci and Tetranacci-type numbers.
Recently, Andrews and EI Bachraoui obtained several iden tities on two-colored partitions. While solving open problems they posed, Chen and Zhou derived a number of identities using analytic methods and asked for combinatorial proofs. In…
We give combinatorial proofs for some identities involving binomial sums that have no closed form.
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.
In this paper we explore two types of tilings of a honeycomb strip and derive some closed form formulas for the number of tilings. Furthermore, we obtain some new identities involving tribonacci numbers, Padovan numbers and Narayana's cow…
We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.
We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, most of them rather involved or else relying on sophisticated number theoretical…
By investigating a recurrence relation about functions, we first give alternative proofs of various identities on Fibonacci numbers and Lucas numbers, and then, make certain well known identities visible via certain trivalent graph…
A number of identities are proved by using Stirling transforms. These identities involve Stirling numbers of the first and second kinds, hyperharmonic and derangement numbers, Bernoulli and Euler numbers and polynomials, powers, power sums,…
We present some new linear, quadratic, cubic and quartic binomial Fibonacci, Lucas and Fibonacci--Lucas summation identities.
We derive some Fibonacci and Lucas identities which contain inverse binomial coefficients. Extension of the results to the general Horadam sequence is possible, in some cases.
We develop the necessary machinery in order to prove that hexagonal tilings are uniquely determined by their Tutte polynomial, showing as an example how to apply this technique to the toroidal hexagonal tiling.
We consider the tiling of an $n$-board (a $1\times n$ array of square cells of unit width) with half-squares ($\frac12\times1$ tiles) and $(\frac12,\frac12)$-fence tiles. A $(\frac12,\frac12)$-fence tile is composed of two half-squares…
The aim of this paper is to present a comprehensive survey of cubic Fibonacci identities, trying to uncover as many as possible. From the outset, our rationale for a very careful search on an apparently obscure problem was not only a matter…