International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 39, Pages 2507-2518
doi:10.1155/S0161171203201071

Netted matrices

Abstract

We prove that powers of 4-netted matrices (the entries satisfy a four-term recurrence δai,j=αai−1,j+βai−1,j+γai,j−1) preserve the property of nettedness: the entries of the eth power satisfy δeai,j(e)=αeai−1,j(e)+βeai−1,j−1(e)+γeai,j−1(e), where the coefficients are all instances of the same sequence xe+1=(β+δ)xe−(βδ+αγ)xe−1. Also, we find a matrix Qn(a,b) and a vector v such that Qn(a,b)e⋅v acts as a shifting on the general second-order recurrence sequence with parameters a, b. The shifting action of Qn(a,b) generalizes the known property (0111)e⋅(1,0)t=(Fe−1,Fe)t. Finally, we prove some results about congruences satisfied by the matrix Qn(a,b).