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Search: id:A072255
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| A072255 |
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Number of ways to partition {1,2,...,n} into arithmetic progressions, where in each partition all the progressions have the same common difference and have lengths greater than or equal to 2. |
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1
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| 1, 1, 3, 4, 7, 11, 19, 29, 47, 76, 125, 200, 322, 519, 845, 1366, 2211, 3573, 5778, 9342, 15122, 24481, 39639, 64094, 103684, 167734, 271397, 439178, 710698
(list; graph; listen)
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OFFSET
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2,3
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REFERENCES
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The question of enumerating these partitions appears as Problem 11005, American Mathematical Monthly, 110, April 2003, page 340.
Problem 11005, American Math. Monthly, Vol. 112, 2005, pp. 89-90. (The published solution is incomplete; the solver's expression q_2(n,d) must be summed over all d = 1,2,...,floor{n/2}.)
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LINKS
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T. D. Noe, Table of n, a(n) for n=2..500
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FORMULA
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a(n) = sum_{d=1}^{floor{n/2}} {{F_k}^r}*{F_{k-1}}^{d-r}, where d is the common difference of the arithmetic progressions, k = Floor{n/d}, r = n mod d and F_k is the k-th Fibonacci number (A000045). - Marty Getz (ffmpg1(AT)uaf.edu) and Dixon Jones (fndjj(AT)uaf.edu), May 21 2005
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EXAMPLE
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a(5)=4: the four ways to partition {1,2,3,4,5} as described above are: {1,2}{3,4,5}; {1,2,3}{4,5}; {1,2,3,4,5}; {1,3,5}{2,4}.
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CROSSREFS
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A053732 relates to partitions of {1, 2, ..., n} into arithmetic progressions without restrictions on the common difference of the progressions.
Sequence in context: A041739 A042593 A041018 this_sequence A049863 A025068 A049928
Adjacent sequences: A072252 A072253 A072254 this_sequence A072256 A072257 A072258
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Marty Getz (ffmpg1(AT)uaf.edu) and Dixon Jones (fndjj(AT)uaf.edu) (fndjj(AT)uaf.edu), Jul 08 2002
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