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.
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved