In this paper, we describe a methodology for mapping normal linear recurrence equations onto a spectrum of systolic architectures. First, we provide a method to map a system of directed recurrence equations, a subclass of linear recurrence equations, onto a very general architecture referred to as basic systolic architecture and establish correctness of the implementation. We also show how eeci...

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We discuss in detail the error bounds for asymptotic solutions of second-order linear difference equation yn 2 n p anyn 1 n q bnyn 0, where p and q are integers, an and bn have asymp...

In this paper we consider second order recurrences {Vk} and {Un} . We give second order linear recurrences for the sequences {V±kn} and {U±kn}. Using these recurrence relations, we derive relationships between the determinants of certain matrices and these sequences. Further, as generalizations of the earlier results, we give representations and trigonometric factorizations of these sequences b...

Fix a prime/?. We say that a set S forms a complete residue system modulop if, for all i such that 0 < i < p 1 , there exists s GS such that s = J (mod /?). We say that a set S forms a reduced residue system modulo p if, for all / such that 1 < i < p -1, there exists s GS such that s = i (mod p). In [9], Shah showed that, ifp is a prime and p = 1,9 (mod 10), then the Fibonacci sequence does not...

We consider functions of natural numbers which allow a combinatorial interpretation as density functions (speed) of classes of relational structures, such as Fibonacci numbers, Bell numbers, Catalan numbers and the like. Many of these functions satisfy a linear recurrence relation over Z or Zm and allow an interpretation as counting the number of relations satisfying a property expressible in M...

A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers {Fn}, where F1 = 1, F2 = 2 and Fn+1 = Fn + Fn−1. For general recurrences {Gn} with non-negative coefficients, there is a notion of a legal decomposition which again leads to a unique representation, and the number of summands in the representations of uni...

We exhibit a three parameter infinite family of quadratic recurrence relations inspired by the well known Somos sequences. For one infinite subfamily we prove that the recurrence generates an infinite sequence of integers by showing that the same sequence is generated by a linear recurrence (with suitable initial conditions). We also give conjectured relations among the three parameters so that...

In this paper we introduce a Ramsey type function S(r; a, b, c) as the maximum s such that for any r-coloring of N there is a monochromatic sequence x1, x2, . . . , xs satisfying a homogeneous second order linear recurrence axi + bxi+1 + cxi+2 = 0, 1 6 i 6 s − 2. We investigate S(2; a, b, c) and evaluate its values for a wide class of triples (a, b, c).

Motivated by the famous Skolem-Mahler-Lech theorem we initiate in this paper the study of a natural class of determinantal varieties which we call Vandermonde varieties. They are closely related to the varieties of linear recurrence relations of a given order possessing a non-trivial solution vanishing at a given set of integers. In the regular case, i.e. when the dimension of a Vandermonde var...