OFFSET
0,2
COMMENTS
Number of representations of n as a sum of six times a square and a square. - Ralf Stephan, May 14 2007
a(n) < 2 if and only if n is in A002480. a(n) > 0 if and only if n is in A002481. - Michael Somos, Mar 01 2011
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, arXiv:math/0611300 [math.NT], 2016-2017.
FORMULA
G.f.: Sum_{ i, j = -oo..+oo } q^(i^2 + 6*j^2).
a(0) = 1, a(n) = (1+(-1)^t)*b(n) for n > 0, where t is the number of prime factors of n, counting multiplicity, which are == 2,3,5,11 (mod 24), and b() is multiplicative with b(p^e) = (e+1) for primes p == 1,5,7,11 (mod 24) and b(p^e) = (1+(-1)^e)/2 for primes p == 13,17,19,23 (mod 24). (This formula is Corollary 4.2 in the Berkovich-Yesilyurt paper). - Jeremy Lovejoy, Nov 14 2024
EXAMPLE
G.f. = 1 + 2*x + 2*x^4 + 2*x^6 + 4*x^7 + 2*x^9 + 4*x^10 + 4*x^15 + 2*x^16 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^6], {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)
PROG
(PARI) {a(n) = my(G); if( n<0, 0, G = [ 1, 0; 0, 6]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, Mar 01 2011 */
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
N. J. A. Sloane, May 18 2002
STATUS
approved