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Search: id:A087153
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| A087153 |
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Number of partitions of n into non-squares. |
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+0
3
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| 0, 1, 1, 1, 2, 3, 3, 5, 5, 8, 9, 13, 15, 20, 24, 30, 37, 47, 55, 71, 83, 103, 123, 151, 178, 218, 257, 310, 366, 440, 515, 617, 722, 857, 1003, 1184, 1380, 1625, 1889, 2214, 2570, 3000, 3472, 4042, 4669, 5414, 6244, 7221, 8303, 9583, 10998, 12655, 14502
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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Also, number of partitions of n where there are fewer than k parts equal to k for all k. - Jon Perry (perry(AT)globalnet.co.uk) and Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 04 2004. E.g. a(8)=5 because we have 8=6+2=5+3=4+4=3+3+
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REFERENCES
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G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 48.
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LINKS
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James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
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FORMULA
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G.f.: Product_{m>0} (1-x^(m^2))/(1-x^m). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 21 2003
a(n) = (1/n)*Sum_{k=1..n} (A000203(k)-A035316(k))*a(n-k), a(0)=1. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 21 2003
G.f.: product(i=1, oo, sum(j=0, i-1, x^(i*j) )). - Jon Perry (perry(AT)globalnet.co.uk), Jul 26 2004
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EXAMPLE
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n=7: 2+5 = 2+2+3 = 7: a(7)=3;
n=8: 2+6 = 2+2+2+2 = 2+3+3 = 3+5 = 8: a(8)=5;
n=9: 2+7 = 2+2+5 = 2+2+2+3 = 3+3+3 = 3+6: a(9)=5.
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MAPLE
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g:=product((1-x^(i^2))/(1-x^i), i=1..70):gser:=series(g, x=0, 60):seq(coeff(gser, x^n), n=1..53); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 09 2006
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MATHEMATICA
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Drop[ CoefficientList[ Series[ Product[ Sum[x^(i*j), {j, 0, i - 1}], {i, 1, 54}], {x, 0, 54}], x], 1] (from Robert G. Wilson v Aug 05 2004)
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CROSSREFS
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Cf. A087154, A001156, A000009, A000037.
Cf. A052335 (<=k parts of k).
Adjacent sequences: A087150 A087151 A087152 this_sequence A087154 A087155 A087156
Sequence in context: A099609 A120249 A058690 this_sequence A134408 A051032 A106530
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KEYWORD
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nonn
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AUTHOR
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Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 21 2003
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