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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Apparently 8 | a(n). - Alexander R. Povolotsky, Oct 01 2011

Examples

			G.f. = 1 + 24*x + 264*x^2 + 1760*x^3 + 7944*x^4 + 25872*x^5 + 64416*x^6 + 133056*x^7 + ...
		

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.
• Index entries for sequences related to sums of squares

Crossrefs

Row d=12 of A122141 and of A319574, 12th column of A286815.

Cf. A000735, A029751.

Programs

Magma

A := Basis( ModularForms( Gamma0(4), 6), 25); A[1] + 24*A[2] + 264*A[3] + 1760*A[4]; /* Michael Somos, Aug 15 2015 */

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 *)

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 */

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