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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| Language: | English |
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Wiley
1991-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171291000625 |
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| author | Clark Kimberling |
| author_facet | Clark Kimberling |
| author_sort | Clark Kimberling |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-378b0232b1eb441ca6fab59bc2885d9a |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1991-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-378b0232b1eb441ca6fab59bc2885d9a2025-08-20T02:07:44ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251991-01-0114345746210.1155/S0161171291000625Partitioning the positive integers with higher order recurrencesClark Kimberling0University of Evansville, 1800 Lincoln Avenue, Evansville 47722, IN, USAAssociated 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.http://dx.doi.org/10.1155/S0161171291000625Stolarsky arraylinear recurrence sequencenearly arithmetic sequence nearly geometric sequence. |
| spellingShingle | Clark Kimberling Partitioning the positive integers with higher order recurrences International Journal of Mathematics and Mathematical Sciences Stolarsky array linear recurrence sequence nearly arithmetic sequence nearly geometric sequence. |
| title | Partitioning the positive integers with higher order recurrences |
| title_full | Partitioning the positive integers with higher order recurrences |
| title_fullStr | Partitioning the positive integers with higher order recurrences |
| title_full_unstemmed | Partitioning the positive integers with higher order recurrences |
| title_short | Partitioning the positive integers with higher order recurrences |
| title_sort | partitioning the positive integers with higher order recurrences |
| topic | Stolarsky array linear recurrence sequence nearly arithmetic sequence nearly geometric sequence. |
| url | http://dx.doi.org/10.1155/S0161171291000625 |
| work_keys_str_mv | AT clarkkimberling partitioningthepositiveintegerswithhigherorderrecurrences |