International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 3, Pages 457-462
doi:10.1155/S0161171291000625

Partitioning the positive integers with higher order recurrences

Abstract

Associated with any irrational number α>1 and the function g(n)=[αn+12] is an array {s(i,j)} of positive integers defined inductively as follows: s(1,1)=1, s(1,j)=g(s(1,j−1)) for all j≥2, s(i,1)= the least positive integer not among s(h,j) for h≤i−1 for i≥2, and s(i,j)=g(s(i,j−1)) for j≥2. This work considers algebraic integers α of degree ≥3 for which the rows of the array s(i,j) partition the set of positive integers. Such an array is called a Stolarsky array. A typical result is the following (Corollary 2): if α is the positive root of xk−xk−1−…−x−1 for k≥3, then s(i,j) is a Stolarsky array.