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A001156
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Number of partitions of n into squares.
(Formerly M0221 N0079)
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110
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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
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OFFSET
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0,5
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COMMENTS
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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, 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 David Hirschhorn, May 05 2005
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REFERENCES
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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).
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LINKS
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J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301.
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Partition
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FORMULA
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G.f.: Product_{m>=1} 1/(1-x^(m^2)).
G.f.: Sum_{n>=0} x^(n^2) / Product_{k=1..n} (1 - x^(k^2)). - Paul D. Hanna, Mar 09 2012
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). - Reinhard Zumkeller, Nov 08 2009
Conjecture (Jan Bohman, Carl-Erik Fröberg, Hans Riesel, 1979): a(n) ~ c * n^(-alfa) * exp(beta*n^(1/3)), where c = 1/18.79656, beta = 3.30716, alfa = 1.16022. - Vaclav Kotesovec, Aug 19 2015
Correct values of these constants are:
1/c = sqrt(3) * (4*Pi)^(7/6) / Zeta(3/2)^(2/3) = 17.49638865935104978665...
alfa = 7/6 = 1.16666666666666666...
beta = 3/2 * (Pi/2)^(1/3) * Zeta(3/2)^(2/3) = 3.307411783596651987...
a(n) ~ 3^(-1/2) * (4*Pi*n)^(-7/6) * Zeta(3/2)^(2/3) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3)). [Hardy & Ramanujan, 1917]
(End)
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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.
G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 +...
A(x) = 1/((1-x)*(1-x^4)*(1-x^9)*(1-x^16)*(1-x^25)*...)
A(x) = 1 + x/(1-x) + x^4/((1-x)*(1-x^4)) + x^9/((1-x)*(1-x^4)*(1-x^9)) + x^16/((1-x)*(1-x^4)*(1-x^9)*(1-x^16)) + ...
The a(14) = 6 integer partitions into squares are:
(941)
(911111)
(44411)
(44111111)
(41111111111)
(11111111111111)
while the a(14) = 6 integer partitions in which the multiplicity of k is a multiple of k for all k are:
(333221)
(33311111)
(22222211)
(2222111111)
(221111111111)
(11111111111111)
(End)
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+ `if`(i^2>n, 0, b(n-i^2, i))))
end:
a:= n-> b(n, isqrt(n)):
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MATHEMATICA
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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}]] (* Robert G. Wilson v, Apr 12 2005, revised Sep 27 2011 *)
f[n_] := Length@ IntegerPartitions[n, All, Range@ Sqrt@ n^2]; Array[f, 67] (* Robert G. Wilson v, Apr 14 2013 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2>n, 0, b[n-i^2, i]]]]; a[n_] := b[n, Sqrt[n]//Floor]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Nov 02 2015, after Alois P. Heinz *)
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PROG
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(Haskell)
a001156 = p (tail a000290_list) where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
(PARI) {a(n)=polcoeff(1/prod(k=1, sqrtint(n+1), 1-x^(k^2)+x*O(x^n)), n)} \\ Paul D. Hanna, Mar 09 2012
(PARI) {a(n)=polcoeff(1+sum(m=1, sqrtint(n+1), x^(m^2)/prod(k=1, m, 1-x^(k^2)+x*O(x^n))), n)} \\ Paul D. Hanna, Mar 09 2012
(Magma) m:=70; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^(k^2)): k in [1..(m+2)]]) )); // G. C. Greubel, Nov 11 2018
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CROSSREFS
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Cf. A001462, A003114, A006141, A011757, A039900, A047993, A052335, A062457, A064174, A078135, A109298, A117144.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Gh. Niculescu (ghniculescu(AT)yahoo.com), Oct 08 2006
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STATUS
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approved
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