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A001156
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Number of partitions of n into squares.
(Formerly M0221 N0079)
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23
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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; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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.rs), 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
| 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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
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LINKS
| 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
| 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.rs), Nov 20 2002
a(n) = f(n,1,3) with f(x,y,z) = if x<y then 0^x else f(x-y,y,z)+f(x,y+z,z+2). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 08 2009]
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EXAMPLE
| 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
| CoefficientList[ Series[Product[1/(1 - x^(m^2)), {m, 70}], {x, 0, 68}], x] (* Or *)
Join[{1}, Table[Length@PowersRepresentations[n, n, 2], {n, 68}]] (* from Robert G. Wilson v, Apr 12 2005, revised Sep 27 2011 *)
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PROG
| (Haskell)
a001156 n = p (map (^ 2) [1..]) n where
p _ 0 = 1
p ks'@(k:ks) m | m < k = 0
| otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 14 2011
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CROSSREFS
| Cf. A131799, A037444, A033461, A000041, A000290.
Sequence in context: A194239 A064475 A025774 * A199119 A035451 A124746
Adjacent sequences: A001153 A001154 A001155 * A001157 A001158 A001159
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| 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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