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 A000145 Number of ways of writing n as a sum of 12 squares. 9
 1, 24, 264, 1760, 7944, 25872, 64416, 133056, 253704, 472760, 825264, 1297056, 1938336, 2963664, 4437312, 6091584, 8118024, 11368368, 15653352, 19822176, 24832944, 32826112, 42517728, 51425088, 61903776, 78146664, 98021616 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Apparently 8 | a(n). - Alexander R. Povolotsky, Oct 01 2011 REFERENCES E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314. LINKS T. D. Noe, Table of n, a(n) for n = 0..10000 Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032. FORMULA Expansion of eta(q^2)^60 / (eta(q) * eta(q^4))^24 in powers of q. Euler transform of period 4 sequence [24, -36, 24, -12, ...]. - Michael Somos, Sep 21 2005 G.f.: (Sum_k x^k^2)^12 = theta_3(q)^12. a(n) = A029751(n) + 16 * A000735(n). - Michael Somos, Sep 21 2005 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). a(n) = (24/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017 EXAMPLE G.f. = 1 + 24*x + 264*x^2 + 1760*x^3 + 7944*x^4 + 25872*x^5 + 64416*x^6 + 133056*x^7 + ... MAPLE (sum(x^(m^2), m=-10..10))^12; # gives g.f. for first 100 terms 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 A000145list := proc(len) series(JacobiTheta3(0, x)^12, x, len+1); seq(coeff(%, x, j), j=0..len-1) end: A000145list(27); # Peter Luschny, Oct 02 2018 MATHEMATICA SquaresR[12, Range[0, 30]] (* Harvey P. Dale, Sep 07 2012 *) a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^12, {q, 0, n}]; (* Michael Somos, Aug 15 2015 *) 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 *) PROG (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 */ (MAGMA) A := Basis( ModularForms( Gamma0(4), 6), 25); A[1] + 24*A[2] + 264*A[3] + 1760*A[4]; /* Michael Somos, Aug 15 2015 */ CROSSREFS Row d=12 of A122141 and of A319574, 12th column of A286815. Cf. A000735, A029751. Sequence in context: A009175 A296916 A187380 * A286346 A126904 A001413 Adjacent sequences:  A000142 A000143 A000144 * A000146 A000147 A000148 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified December 9 17:51 EST 2018. Contains 318023 sequences. (Running on oeis4.)