Integer compositions with certain colored parts were introduced by Andrews in 2007 to address a number-theoretic problem. allowing zero as some Ouvry and Polychronakos 2019. We give bijection between these two varieties of determine various combinatorial properties multicompositions. In particular, we the count multicompositions number all parts, positive zeros. Then, working from three types r...

In this study, we define a new type of Pell and Pell-Lucas numbers which are called Gaussian-bihyperbolic numbers. We also negaGaussian-bihyperbolic Moreover, obtain Binet’s formulas, generating function d’Ocagne’s identities, Catalan’s Cassini’s identities some sum formulas for these investigate algebraic proporties these. Furthermore, give the matrix representation

In this paper, we introduce the incomplete Vieta–Pell and Vieta–Pell–Lucas polynomials. We give some properties, recurrence relations generating function of these polynomials with suggestions for further research.

Abstract By means of the telescoping method, several summation formulae are established for arctangent function with its argument being Pell and Pell–Lucas polynomials. Numerous infinite series identities involving Fibonacci Lucas numbers included as particular cases.

An integer sequence a(n) is called a jump sequence if a(1) = 1 and 1 ≤ a(n) < n for n ≥ 2. Such a sequence has the property that ak(n) = a(a(· · · (a(n)) · · · )) goes to 1 in finitely many steps. We call the pattern (n, a(n), a2(n), . . . , al(n) = 1) a jumping pattern from n down to 1. In this paper we look at jumping sequences that are weight minimizing with respect to various weight functio...

We describe the recursive structures of the set of two-stack sortable permutations which avoid 132 and the set of two-stack sortable permutations which contain 132 exactly once. Using these results and standard generating function techniques, we enumerate two-stack sortable permutations which avoid (or contain exactly once) 132 and which avoid (or contain exactly once) an arbitrary permutation ...