would you be inclined to try to prove these by counting arguments? Probably not. We confess that there was a time that when we saw alternating sum identities like the ones above, we would attack it with noncombinatorial proof techniques, like induction or generating functions. After all, how can an object be added a negative number of times? Perhaps it can be tackled using the Principle of Incl...

In the paper, the authors establish two identities to express higher order derivatives and integer powers of the generating function of the Chebyshev polynomials of the second kind in terms of integer powers and higher order derivatives of the generating function of the Chebyshev polynomials of the second kind respectively, find an explicit formula and an identity for the Chebyshev polynomials ...

We consider a weighted square-and-domino tiling model obtained by assigning real number weights to the cells and boundaries of an n-board. An important special case apparently arises when these weights form periodic sequences. When the weights of an nm-tiling form sequences having period m, it is shown that such a tiling may be regarded as a meta-tiling of length n whose weights have period 1 e...

In this paper, we consider a generalized Catalan triangle de…ned by km n 2n n k for positive integer m: Then we compute the weighted half binomial sums with the certain powers of generalized Fibonacci and Lucas numbers of the form n X k=0 2n n+ k km n X tk; where Xn either generalized Fibonacci or Lucas numbers, t and r are integers for 1 m 6: After we describe a general methodology to show how...

We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves combinatorics and formal Laurent series. Using very little analysis we can express their limits in terms of (modified) Bessel functions and Fibonacci polynomials....

In this paper, we introduce tiling representations of Fibonacci p-numbers, which are generalizations the well-known and Narayana numbers, generalized in distance sense. We obtain p-numbers count number distinct ways to tile a 1 × n board using various r, r-ominoes from r = up p + 1. Moreover, product identities sum formulas these numbers with special subscripts given by interpretations that all...

a non-abelian finite group is called sequenceable if for some positive integer , is -generated ( ) and there exist integers such that every element of is a term of the -step generalized fibonacci sequence , , , . a remarkable application of this definition may be find on the study of random covers in the cryptography. the 2-step generalized sequences for the dihedral groups studied for their pe...

A graph is said to be graded if its vertices are divided into levels numbered by integers, so that the endpoints of any edge lie on consecutive levels. Discrete modular lattices and rooted trees are among the typical examples. The following three types of problems are of interest to us: (1) path counting in graded graphs, and related combinatorial identities; (2) bijective proofs of these ident...