The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A033712 theta3(z) * theta3(2*z) * theta3(3*z) * theta3(6*z). 12
 1, 2, 2, 6, 6, 4, 14, 8, 6, 26, 12, 16, 42, 12, 16, 44, 6, 20, 50, 16, 36, 56, 24, 16, 42, 30, 28, 78, 48, 36, 84, 40, 6, 80, 36, 48, 150, 44, 40, 100, 36, 36, 112, 48, 72, 148, 48, 48, 42, 50, 62, 124, 84, 52, 158, 64, 48, 144, 60, 64, 252, 60, 64, 200, 6, 88, 168, 64, 108 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). REFERENCES J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102, eq. 9. L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 225. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Number of solutions to a^2 + 2*b^2 + 3*c^2 + 6*d^2 = n in integers. Expansion of phi(q) * phi(q^2) * phi(q^3) * phi(q^6) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Apr 19 2015 Expansion of (eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12))^3 / (eta(q) * eta(q^3) * eta(q^8) * eta(q^24))^2 in powers of q. Euler transform of period 24 sequence [2, -1, 4, -4, 2, -2, 2, -2, 4, -1, 2, -8, 2, -1, 4, -2, 2, -2, 2, -4, 4, -1, 2, -4, ...]. - Michael Somos, May 30 2005 G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Apr 19 2015 a(2*n) = A282544(n). a(4*n) = A125510(n). EXAMPLE G.f. = 1 + 2*q + 2*q^2 + 6*q^3 + 6*q^4 + 4*q^5 + 14*q^6 + 8*q^7 + 6*q^8 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3] EllipticTheta[ 3, 0, q^6], {q, 0, n}]; (* Michael Somos, Apr 19 2015 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A * subst( A + x * O(x^(n\2)), x , x^2) * subst( A + x * O(x^(n\3)), x, x^3) * subst( A + x * O(x^(n\6)), x, x^6), n))}; /* Michael Somos, May 30 2005 */ (PARI) {a(n) = my(G); if( n<0, 0, G = [1, 0, 0, 0; 0, 2, 0, 0; 0, 0, 3, 0; 0, 0, 0, 6 ]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, Apr 19 2015 */ (Magma) A := Basis( ModularForms( Gamma0(24), 2), 69); A[1] + 2*A[2] + 2*A[3] + 6*A[4] + 6*A[5] + 4*A[6] + 14*A[7] + 6*A[8]; /* Michael Somos, Apr 19 2015 */ CROSSREFS Cf. A125510, A282544. Sequence in context: A320190 A320189 A320188 * A033730 A033754 A171661 Adjacent sequences: A033709 A033710 A033711 * A033713 A033714 A033715 KEYWORD nonn AUTHOR N. J. A. Sloane STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 1 08:00 EST 2024. Contains 370430 sequences. (Running on oeis4.)