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Search: id:A001156
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| A001156 |
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Number of partitions of n into squares.
(Formerly M0221 N0079)
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+0
18
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| 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 8, 9, 10, 10, 12, 13, 14, 14, 16, 19, 20, 21, 23, 26, 27, 28, 31, 34, 37, 38, 43, 46, 49, 50, 55, 60, 63, 66, 71, 78, 81, 84, 90, 98, 104, 107, 116, 124, 132, 135, 144, 154, 163, 169, 178, 192, 201, 209, 220, 235, 247, 256
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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Number of partitions of n such that number of parts equal to k is multiple of k for all k. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 01 2004
Of course p_{4*square}(n)>0. In fact p_{4*square}(32n+28)=3 times p_{4*square}(8n+7) and p_{4*square}(72n+69) is even. These seem to be the only arithmetic properties the function p_{4*square(n)} possesses. Similar results hold for partitions into positive squares, distinct squares and distinct positive squares. - Michael D. Hirschhorn (m.hirschhorn(AT)unsw.edu.au), May 05 2005
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REFERENCES
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J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301.
M. D. Hirschhorn and J. A. Sellers, On a problem of Lehmer on partitions into squares, The Ramanujan Journal 8 (2004), 279-287.
F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Science, Vol. 16E, No. 2 (1997), pp. 237-240.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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G.f.: Product 1/(1-x^(m^2)); m=1..inf.
a(n) = 1/n*Sum_{k=1..n} A035316(k)*a(n-k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Nov 20 2002
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EXAMPLE
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p_{4*square}(23)=1 because 23=3^2+3^2+2^2+1^2, and there is no other partition of 23 into squares.
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MATHEMATICA
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CoefficientList[ Series[Product[1/(1 - x^(m^2)), {m, 70}], {x, 0, 68}], x] (* Or *)
(* first do *) Needs["NumberTheory`NumberTheoryFunctions`"] (* then *) Table[ Length @ SumOfSquaresRepresentations[n, n], {n, 68}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 12 2005)
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CROSSREFS
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Cf. A131799.
Adjacent sequences: A001153 A001154 A001155 this_sequence A001157 A001158 A001159
Sequence in context: A064775 A064475 A025774 this_sequence A035451 A124746 A124789
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KEYWORD
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nonn,easy
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AUTHOR
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njas
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EXTENSIONS
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More terms from Eric Weisstein (eric(AT)weisstein.com)
More terms from Gh. Niculescu (ghniculescu(AT)yahoo.com), Oct 08 2006
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