International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 3, Pages 457-462

Partitioning the positive integers with higher order recurrences

Clark Kimberling

University of Evansville, 1800 Lincoln Avenue, Evansville 47722, IN, USA

Received 17 June 1990; Revised 25 January 1991

Copyright © 1991 Clark Kimberling. 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.


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,j1)) for all j2, s(i,1)= the least positive integer not among s(h,j) for hi1 for i2, and s(i,j)=g(s(i,j1)) for j2. 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 xkxk1x1 for k3, then s(i,j) is a Stolarsky array.