This article describes conjectured combinatorial interpretations for the higher q, t-Catalan sequences introduced by Garsia and Haiman, which arise in the theory of symmetric functions and Macdonald polynomials. We define new combinatorial statistics generalizing those proposed by Haglund and Haiman for the original q, tCatalan sequence. We prove explicit summation formulas, bijections, and rec...

In this paper, given an integer e and n such that e|n, and a prime p, we propose a method of constructing optimal p2-ary low correlation zone (LCZ) sequence set with parameters (pn−1, pe−1, (pn −1)/(pe − 1), 1) from a p-ary sequence of the same length with ideal autocorrelation. The resulting p2-ary LCZ sequence set can be viewed as the generalization of the optimal quaternary LCZ sequence set ...

We show that there are relationships between a generalized Lucas sequence and the permanent and determinant of some Hessenberg matrices.

A helpful starting point is the paper entitled "Fibonacci Series Modulo m by D. D. Wall [3]. With Wall, we let /„ denote the n member of the sequence of integers fQ = a, f1 = b9 ..., where fn + 1 = fn + f n _ r The symbol h(m) will denote the length of the period of the sequence resulting from reducing each /„ modulo w. The basic Fibonacci sequence will be given by uQ = 0, ux = 1, ... and the L...

Let P be an odd integer, (Un) and (Vn) denote generalized Fibonacci and Lucas sequences defined by U0 = 0, U1 = 1, and Un+1 = PUn +Un 1, V0 = 2, V1 = P, and Vn+1 = PVn + Vn 1 for n 1. In this paper, we solve the equations Un = kx2 ± 1 under some conditions on n. Moreover, we determine all indices n such that the equations Vn = wkx2 ± 1, where w 2 {1, 2, 3, 6} , k|P with k > 1, have solutions.

In the present paper, we study the class LPn which consists of Laurent polynomials P (z) = (n+ 1) + n ∑ k=1 k – odd ck(z k + z−k), with all coefficients ck equal to −1 or 1. Such polynomials arise in the study of Barker sequences of even length — binary sequences with minimal possible autocorrelations. By using an elementary (but not trivial) analytic argument, we prove that polynomials Rn(z) w...

In this paper we obtain some congruences involving central binomial coefficients and Lucas sequences. For example, we show that if p > 5 is a prime then p−1

In this paper, we look at those pairs of integers (a, b) for which the Lucas sequence of general term un = un(a, b) has the property that 4 | φ(|un|) for almost all positive integers n.

We give the bicomplex Gaussian Fibonacci and Lucas numbers establish generating functions Binet’s formulas related to these numbers. Also, we present summation formula, matrix representation Honsberger identity their relationship between Finally, show relationships among Fibonacci, Lucas,

Near normal sequences of new lengths n = 4m + 1 = 49,53,57 are constructed. The relation between a special set of near normal sequences and Golay sequences is discussed. A reformulation of Yang's powerful theorems on T-sequences is also given. We give base sequences for lengths m + p, m p, m, m for p 1 and m E {19, ... , 30}. Some of these are new lengths, or new decompositions into four square...