A123063 Theta series of lattice with Gram matrix [4,1;1,8].

1, 0, 2, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 4, 0, 2, 2, 4, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 4, 0, 0, 2, 2, 0, 2, 0, 6, 2, 0, 0, 0, 2, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 6, 0, 0, 2, 0, 0, 2, 2, 4, 0, 0, 0, 0, 0, 6, 2, 2, 0, 0, 0, 4, 0, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 2, 0, 2, 4, 0, 6, 2, 0, 2, 0

Offset

0,3

Comments

a(n) = number of solutions to n = 2*x^2 + x*y + 4*y^2 in integers, hence a(n) nonzero if and only if n is in A123064 and p is prime and a(p) = 2 if and only if p is in A106872. - Michael Somos, Jul 16 2011

References

J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 82.

Links

Table of n, a(n) for n=0..104.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Formula

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u6*u1^3 + 2*u3*u2^3 - 3*u3^3*u2 - 6*u6^3*u1 + 6*u6*u2^2*u1 - 6*u3*u2^2*u1 + 3*u3*u2*u1^2 - 6*u6*u2*u1^2 - 9*u6*u3^2*u1 - 18*u6^2*u3*u2 + 18*u6*u3^2*u2 + 18*u6^2*u3*u1. - Michael Somos, Sep 28 2006

G.f. is a period 1 Fourier series which satisfies f(-1 / (31 t)) = 31^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 16 2011

G.f.: Sum_{n,m in Z} x^(2*n^2 + n*m + 4*m^2).

Example

G.f. = 1 + 2*x^2 + 2*x^4 + 2*x^5 + 2*x^7 + 2*x^8 + 2*x^10 + 2*x^14 + 4*x^16 + 2*x^18 + ...
G.f. =  1 + 2*q^4 + 2*q^8 + 2*q^10 + 2*q^14 + 2*q^16 + 2*q^20 + 2*q^28 + 4*q^32 + ...

Mathematica

terms = 105; max = terms+3; s = Sum[x^(2*n^2 + n*m + 4*m^2), {n, -max, max}, {m, -max, max}] + O[x]^max; CoefficientList[s, x][[1 ;; terms]] (* Jean-François Alcover, Jul 05 2017 *)

Prog

(Magma) L:=LatticeWithGram(2, [4, 1, 1, 8] ); T<q> := ThetaSeries(L, 500); T;
(PARI) {a(n) = if( n<1, n==0, qfrep( [4, 1; 1, 8], n, 1)[n] * 2)}; /* Michael Somos, Sep 28 2006 */
(Magma) A := Basis( ModularForms( Gamma1(31), 1), 103); A[1] + 2*A[3] + 2*A[5] + 2*A[6] + 2*A[8] + 2*A[9] + 2*A[11] + 2*A[15]; /* Michael Somos, Jun 14 2014 */
(Magma) a := func <n | Coefficient( ThetaSeries( LatticeWithGram( 2, [4, 1, 1, 8]), 2*n), 2*n)>; /* Michael Somos, Jun 14 2014 */

Crossrefs

Cf. A106872, A123064, A123065.

Sequence in context: A161516 A347730 A329491 * A031358 A279103 A318734

Adjacent sequences: A123060 A123061 A123062 * A123064 A123065 A123066

Keyword

nonn

Author

N. J. A. Sloane, Sep 27 2006

Status

approved