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%I #29 Sep 08 2022 08:45:28
%S 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,
%T 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,
%U 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
%N Theta series of lattice with Gram matrix [4,1;1,8].
%C 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
%D J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 82.
%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%F 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
%F 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
%F G.f.: Sum_{n,m in Z} x^(2*n^2 + n*m + 4*m^2).
%e 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 + ...
%e 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 + ...
%t 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 *)
%o (Magma) L:=LatticeWithGram(2, [4,1,1,8] ); T<q> := ThetaSeries(L,500); T;
%o (PARI) {a(n) = if( n<1, n==0, qfrep( [4, 1; 1, 8], n, 1)[n] * 2)}; /* _Michael Somos_, Sep 28 2006 */
%o (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 */
%o (Magma) a := func <n | Coefficient( ThetaSeries( LatticeWithGram( 2, [4, 1, 1, 8]), 2*n), 2*n)>; /* _Michael Somos_, Jun 14 2014 */
%Y Cf. A106872, A123064, A123065.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Sep 27 2006
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