OFFSET
0,2
COMMENTS
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102, eq. 9.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 225.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Number of solutions to a^2 + 2*b^2 + 3*c^2 + 6*d^2 = n in integers.
Expansion of phi(q) * phi(q^2) * phi(q^3) * phi(q^6) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Apr 19 2015
Expansion of (eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12))^3 / (eta(q) * eta(q^3) * eta(q^8) * eta(q^24))^2 in powers of q.
Euler transform of period 24 sequence [2, -1, 4, -4, 2, -2, 2, -2, 4, -1, 2, -8, 2, -1, 4, -2, 2, -2, 2, -4, 4, -1, 2, -4, ...]. - Michael Somos, May 30 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Apr 19 2015
EXAMPLE
G.f. = 1 + 2*q + 2*q^2 + 6*q^3 + 6*q^4 + 4*q^5 + 14*q^6 + 8*q^7 + 6*q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3] EllipticTheta[ 3, 0, q^6], {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A * subst( A + x * O(x^(n\2)), x , x^2) * subst( A + x * O(x^(n\3)), x, x^3) * subst( A + x * O(x^(n\6)), x, x^6), n))}; /* Michael Somos, May 30 2005 */
(PARI) {a(n) = my(G); if( n<0, 0, G = [1, 0, 0, 0; 0, 2, 0, 0; 0, 0, 3, 0; 0, 0, 0, 6 ]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, Apr 19 2015 */
(Magma) A := Basis( ModularForms( Gamma0(24), 2), 69); A[1] + 2*A[2] + 2*A[3] + 6*A[4] + 6*A[5] + 4*A[6] + 14*A[7] + 6*A[8]; /* Michael Somos, Apr 19 2015 */
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved