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 A123063 Theta series of lattice with Gram matrix [4,1;1,8]. 4
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 := 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 ; /* 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

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Last modified September 22 17:39 EDT 2021. Contains 347607 sequences. (Running on oeis4.)