Partitioning the positive integers with higher order recurrences
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 w...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1991-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171291000625 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |