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A000164
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Number of partitions of n into 3 squares (allowing part zero).
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7
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1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 1, 1, 0, 1, 2, 2, 2, 0, 2, 1, 0, 1, 2, 2, 1, 2, 1, 2, 0, 1, 3, 1, 1, 1, 2, 1, 0, 1, 2, 3, 2, 1, 2, 3, 0, 1, 2, 1, 2, 0, 2, 2, 0, 1, 3, 3, 1, 2, 2, 1, 0, 2, 2, 3, 2, 1, 2, 1, 0, 1, 4, 2, 2, 1, 2, 3, 0, 1, 4, 3, 1, 0, 1, 2, 0, 1, 2, 3, 3, 2, 4, 2, 0, 2
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OFFSET
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0,10
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COMMENTS
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a(n) = number of triples of integers [ i, j, k] such that i >= j >= k >= 0 and n = i^2 + j^2 + k^2. - Michael Somos, Jun 05 2012
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REFERENCES
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E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 84.
Hirschhorn, M. D.; Some formulae for partitions into squares, DiscreteMath, 211 (2000), pp. 225-228. [From Ant King, Oct 15 2010]
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..10000
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FORMULA
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Let e(n,r,s,m) be the excess of the number of n's r(mod m) divisors over the number of its s(mod m) divisors, and let delta(n)=1 if n is a perfect square and 0 otherwise. Then, if we define alpha(n)=5delta(n)+3 delta(1/2 n)+ 4delta(1/3 n), beta(n)=4e(n,1,3,4)+3e(n,1,7,8)+3e(n,3,5,8), gamma(n)=2 sum(e(n-k^2,1,3,4),1<=k^2<n), it follows that a(n)=1/12 (alpha(n)+beta(n)+gamma(n)) - Ant King, Oct 15 2010]
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EXAMPLE
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1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^8 + 2*x^9 + x^10 + x^11 + x^12 + x^13 + ...
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MATHEMATICA
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Length[PowersRepresentations[ #, 3, 2]] & /@ Range[0, 104]
e[0, r_, s_, m_]=0; e[n_, r_, s_, m_]:=Length[Select[Divisors[n], Mod[ #, m]==r &]]-Length[Select[Divisors[n], Mod[ #, m]==s &]]; alpha[n_]:=5delta[n]+3delta[1/2 n]+4delta[1/3n]; beta[n_]:=4e[n, 1, 3, 4]+3e[n, 1, 7, 8]+3e[n, 3, 5, 8]; delta[n_]:=If[IntegerQ[Sqrt[n]], 1, 0]; f[n_]:=Table[n-k^2, {k, 1, Floor[Sqrt[n]]}]; gamma[n_]:=2 Plus@@(e[ #, 1, 3, 4] &/@f[n]); p3[n_]:=1/12(alpha[n]+beta[n]+gamma[n]); p3[ # ] &/@Range[0, 104]
(* Ant King, Oct 15 2010 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, sum( i=0, sqrtint(n), sum( j=0, i, sum( k=0, j, n == i^2 + j^2 + k^2))))} /* Michael Somos, Jun 05 2012 */
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CROSSREFS
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Sequence in context: A169987 A178666 A206706 * A157746 A037820 A076493
Adjacent sequences: A000161 A000162 A000163 * A000165 A000166 A000167
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Name clarified. - Wolfdieter Lang, Apr 08 2013
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STATUS
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approved
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