Partitioning the positive integers with higher order recurrences

Clark Kimberling

International Journal of Mathematics and Mathematical Sciences, 1991, vol. 14, 1-6

Abstract:

Associated with any irrational number α > 1 and the function g ( n ) = [ α n + 1 2 ] 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 x k − x k − 1 − … − x − 1 for k ≥ 3 , then s ( i , j ) is a Stolarsky array.

Date: 1991
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:721296

DOI: 10.1155/S0161171291000625

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