Search: id:A001156
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%I A001156 M0221 N0079
%S A001156 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,
%T A001156 26,27,28,31,34,37,38,43,46,49,50,55,60,63,66,71,78,81,84,90,98,104,
%U A001156 107,116,124,132,135,144,154,163,169,178,192,201,209,220,235,247,256
%N A001156 Number of partitions of n into squares.
%C A001156 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
%C A001156 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
%D A001156 J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling 
               (BIT) 19 (1979), 297-301.
%D A001156 M. D. Hirschhorn and J. A. Sellers, On a problem of Lehmer on partitions 
               into squares, The Ramanujan Journal 8 (2004), 279-287.
%D A001156 F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure 
               Appl. Science, Vol. 16E, No. 2 (1997), pp. 237-240.
%D A001156 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001156 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001156 F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, 
               Phoenix, 2006.
%H A001156 T. D. Noe, <a href="b001156.txt">Table of n, a(n) for n=0..1000</a>
%H A001156 James A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">
               Partitions Excluding Specific Polygonal Numbers As Parts</a>, Journal 
               of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
%H A001156 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Sequences-book.pdf">
               Sequences of Numbers Involved in Unsolved Problems</a>.
%H A001156 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Partition.html">Link to a section of The World of Mathematics.</a>
%H A001156 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SmarandacheSequences.html">Link to a section of The World of Mathematics.</
               a>
%H A001156 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SquareNumber.html">Link to a section of The World of Mathematics.</
               a>
%F A001156 G.f.: Product 1/(1-x^(m^2)); m=1..inf.
%F A001156 a(n) = 1/n*Sum_{k=1..n} A035316(k)*a(n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Nov 20 2002
%F A001156 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]
%e A001156 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.
%t A001156 CoefficientList[ Series[Product[1/(1 - x^(m^2)), {m, 70}], {x, 0, 68}], 
               x] (* Or *)
%t A001156 (* 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)
%Y A001156 Cf. A131799.
%Y A001156 Sequence in context: A064775 A064475 A025774 this_sequence A035451 A124746 
               A124789
%Y A001156 Adjacent sequences: A001153 A001154 A001155 this_sequence A001157 A001158 
               A001159
%K A001156 nonn,easy
%O A001156 0,5
%A A001156 N. J. A. Sloane (njas(AT)research.att.com).
%E A001156 More terms from Eric Weisstein (eric(AT)weisstein.com)
%E A001156 More terms from Gh. Niculescu (ghniculescu(AT)yahoo.com), Oct 08 2006

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