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A000145 Number of ways of writing n as a sum of 12 squares.

%I A000145 #55 Sep 08 2022 08:44:27
%S A000145 1,24,264,1760,7944,25872,64416,133056,253704,472760,825264,1297056,
%T A000145 1938336,2963664,4437312,6091584,8118024,11368368,15653352,19822176,
%U A000145 24832944,32826112,42517728,51425088,61903776,78146664,98021616
%N A000145 Number of ways of writing n as a sum of 12 squares.
%C A000145 Apparently 8 | a(n). - _Alexander R. Povolotsky_, Oct 01 2011
%D A000145 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
%D A000145 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.
%H A000145 T. D. Noe, <a href="/A000145/b000145.txt">Table of n, a(n) for n = 0..10000</a>
%H A000145 Shi-Chao Chen, <a href="http://dx.doi.org/10.1016/j.jnt.2010.01.011">Congruences for rs(n)</a>, Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
%H A000145 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A000145 Expansion of eta(q^2)^60 / (eta(q) * eta(q^4))^24 in powers of q.
%F A000145 Euler transform of period 4 sequence [24, -36, 24, -12, ...]. - _Michael Somos_, Sep 21 2005
%F A000145 G.f.: (Sum_k x^k^2)^12 = theta_3(q)^12.
%F A000145 a(n) = A029751(n) + 16 * A000735(n). - _Michael Somos_, Sep 21 2005
%F A000145 G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 64 (t/i)^6 f(t) where q = exp(2 Pi i t).
%F A000145 a(n) = (24/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, May 27 2017
%e A000145 G.f. = 1 + 24*x + 264*x^2 + 1760*x^3 + 7944*x^4 + 25872*x^5 + 64416*x^6 + 133056*x^7 + ...
%p A000145 (sum(x^(m^2),m=-10..10))^12; # gives g.f. for first 100 terms
%p A000145 t1:=(sum(x^(m^2), m=-n..n))^12; t2:=series(t1,x,n+1); t2[n+1]; # _N. J. A. Sloane_, Oct 01 2011
%p A000145 A000145list := proc(len) series(JacobiTheta3(0, x)^12, x, len+1);
%p A000145 seq(coeff(%, x, j), j=0..len-1) end: A000145list(27); # _Peter Luschny_, Oct 02 2018
%t A000145 SquaresR[12,Range[0,30]] (* _Harvey P. Dale_, Sep 07 2012 *)
%t A000145 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^12, {q, 0, n}]; (* _Michael Somos_, Aug 15 2015 *)
%t A000145 nmax = 30; CoefficientList[Series[Product[(1 - x^(2*k))^12 * (1 + x^(2*k - 1))^24, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 10 2018 *)
%o A000145 (PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^12, n))}; /* _Michael Somos_, Sep 21 2005 */
%o A000145 (Magma) A := Basis( ModularForms( Gamma0(4), 6), 25); A[1] + 24*A[2] + 264*A[3] + 1760*A[4]; /* _Michael Somos_, Aug 15 2015 */
%Y A000145 Row d=12 of A122141 and of A319574, 12th column of A286815.
%Y A000145 Cf. A000735, A029751.
%K A000145 nonn,easy
%O A000145 0,2
%A A000145 _N. J. A. Sloane_