OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Theta series of quadratic form (or lattice) with Gram matrix [ 2, 1; 1, 8 ].
The number of integer solutions (x, y) to x^2 + x*y + 4*y^2 = n, discriminant -15. - Ray Chandler, Jul 12 2014
a(n) = number of solutions in integers (x, y) of x^2 + 15*y^2 = 4*n. - Michael Somos, Jul 17 2018
LINKS
John Cannon, Table of n, a(n) for n = 0..5000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of phi(q) * phi(q^15) + 4 * q^4 * psi(q^2) * psi(q^30) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 26 2006
Expansion of (eta(q^3) * eta(q^5))^2 / (eta(q)* eta(q^15)) + (eta(q) *eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q. - Michael Somos, Aug 26 2006
a(n) = A260671(4*n). - Michael Somos, Jul 17 2018
EXAMPLE
G.f. = 1 + 2*q^2 + 6*q^8 + 4*q^12 + 2*q^18 + 4*q^20 + 2*q^30 + 10*q^32 + 4*q^38 + 8*q^48 + 2*q^50 + 4*q^62 + 8*q^68 + 6*q^72 + 8*q^80 + 8*q^92 + 2*q^98 + ...
G.f. = 1 + 2*x + 6*x^4 + 4*x^6 + 2*x^9 + 4*x^10 + 2*x^15 + 10*x^16 + 4*x^19 + ... - Michael Somos, Jul 17 2018
MATHEMATICA
r[n_] := Reduce[x^2 + x*y + 4*y^2 == n, {x, y}, Integers]; Table[rn = r[n]; Which[rn === False, 0, Head[rn] === Or, Length[rn], Head[rn] === And, 1], {n, 0, 105}] (* Jean-François Alcover, Nov 05 2015, after the comment by Ray Chandler *)
a[0] = 1; a[n_] := With[{K = KroneckerSymbol}, DivisorSum[n, K[-15, #] + K[#, 3]*K[n/#, 5]&]]; Table[a[n], {n, 0, 103}] (* Jean-François Alcover, Jul 07 2017, after Michael Somos *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^15] + EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^15], {x, 0, n}]; (* Michael Somos, Jul 17 2018 *)
PROG
(PARI) {a(n) = if( n<1, n==0, qfrep([2, 1; 1, 8], n, 1)[n]*2)}; /* Michael Somos, Aug 26 2006 */
(PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-15, d) + kronecker(d, 3) * kronecker(n/d, 5) ))}; /* Michael Somos, Aug 26 2006 */
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved