cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A033717 Number of integer solutions to the equation x^2 + 2*y^2 + 4*z^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 8, 8, 6, 6, 8, 4, 8, 12, 0, 8, 12, 8, 10, 12, 8, 8, 24, 8, 8, 14, 8, 16, 16, 4, 0, 16, 6, 16, 16, 8, 12, 20, 24, 8, 24, 8, 16, 20, 8, 20, 0, 16, 24, 18, 10, 8, 24, 12, 32, 24, 0, 16, 24, 12, 16, 20, 0, 24, 12, 8, 16, 28, 16, 16, 48, 8, 30, 32, 8, 20, 24, 16, 0, 16, 24, 18
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 2*q + 2*q^2 + 4*q^3 + 4*q^4 + 4*q^5 + 8*q^6 + 8*q^7 + 6*q^8 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.

Links

Crossrefs

Cf. A004015, A008443, A014455, A033763, A045826, A045828, A045831, A045834, A213022, A213622, A213624, A213625, A246811.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(16), 3/2), 82); A[1] + 2*A[2] + 2*A[3] + 4*A[4] + 4*A[5] + 4*A[6] + 8*A[7] + 8*A[8] + 6*A[9] + 8*A[10] + 4*A[11]; /* Michael Somos, Sep 03 2014 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^4], {q, 0, n}]; (* Michael Somos, Sep 03 2014 *)
  • PARI
    {a(n) = my(G); if( n<0, 0, G = [1, 0, 0; 0, 2, 0; 0, 0, 4]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n)), n))}; /* Michael Somos, Sep 03 2014 */
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A) * eta(x^8 + A)^3 / (eta(x + A)^2 * eta(x^16 + A)^2), n))}; /* Michael Somos, Sep 03 2014 */

Formula

Expansion of phi(q) * phi(q^2) * phi(q^4) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Sep 03 2014
Euler transform of period 16 sequence [2, -1, 2, -2, 2, -1, 2, -5, 2, -1, 2, -2, 2, -1, 2, -3, ...]. - Michael Somos, Sep 03 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 03 2014
a(2*n + 1) = 2 * A045828(n). a(4*n) = A014455(n). a(4*n + 1) = 2 * A213625(n). a(4*n + 2) = 2 * A246811(n). a(4*n + 3) = 4 * A213624(n). - Michael Somos, Sep 03 2014
a(8*n) = A005875(n). a(8*n + 1) = 2 * A213622(n). a(8*n + 2) = 2 * A045834(n). a(8*n + 7) = 8 * A033763(n). - Michael Somos, Sep 03 2014
a(16*n) = A004015(n). a(16*n + 2) = 2 * A213022(n). a(16*n + 6) = 8 *
A008443(n). a(16*n + 8) = 2 * A045826(n). a(16*n + 10) = 8 * A045831(n). a(16*n + 14) = 0. - Michael Somos, Sep 03 2014
G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4).

A246631 Number of integer solutions to x^2 + 2*y^2 + 2*z^2 = n.

Original entry on oeis.org

1, 2, 4, 8, 6, 8, 8, 0, 12, 10, 8, 24, 8, 8, 16, 0, 6, 16, 12, 24, 24, 16, 8, 0, 24, 10, 24, 32, 0, 24, 16, 0, 12, 16, 16, 48, 30, 8, 24, 0, 24, 32, 16, 24, 24, 24, 16, 0, 8, 18, 28, 48, 24, 24, 32, 0, 48, 16, 8, 72, 0, 24, 32, 0, 6, 32, 32, 24, 48, 32, 16, 0
Offset: 0

Views

Author

Michael Somos, Aug 31 2014

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 6*q^4 + 8*q^5 + 8*q^6 + 12*q^8 + 10*q^9 + ...

Links

Crossrefs

Cf. A004015, A005875, A008443, A014455, A033717, A045826, A045828, A045831, A045834, A212885, A213022, A213624, A213625, A246811.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(8), 3/2), 80); A[1] + 2*A[2];
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2]^2, {q, 0, n}];
  • PARI
    {a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0, 0; 0, 2, 0; 0, 0, 2], n)[n])};
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^8 / (eta(x + A)^2 * eta(x^8 + A)^4), n))};

Formula

Theta series of quadratic form with Gram matrix [ 1, 0, 0; 0, 2, 0; 0, 0, 2 ].
Expansion of phi(q) * phi(q^2)^2 = phi(-q^4)^4 / phi(-q) in powers of q where phi() is a Ramanujan theta function.
Expansion of eta(q^2) * eta(q^4)^8 / (eta(q)^2 * eta(q^8)^4) in powers of q.
Euler transform of period 8 sequence [ 2, 1, 2, -7, 2, 1, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 4 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A014455.
G.f.: theta_3(q) * theta_3(q^2)^2.
G.f.: Product{k>0} (1 - x^(2*k)) * (1 - x^(4*k))^8 / ((1 - x^k)^2 * (1 - x^(8*k))^4).
G.f.: Product{k>0} (1 + x^(2*k)) * (1 + x^k)^2 * (1 - x^(4*k))^3 / (1 + x^(4*k))^4.
a(n) = (-1)^floor((n+1) / 2) * A212885(n) = abs(A212885(n)).
a(n) = A033717(2*n). a(2*n) = A014455(n). a(2*n + 1) = 2 * A246811(n).
a(4*n) = A005875(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = 4 * A045828(n).
a(8*n) = A004015(n). a(8*n + 1) = 2 * A213022(n). a(8*n + 2) = 4 * A213625(n). a(8*n + 3) = 8 * A008443(n). a(8*n + 4) = 2 * A045826(n). a(8*n + 5) = 8 * A045831(n). a(8*n + 6) = 8 * A213624(n). a(8*n + 7) = 0.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.