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.

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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.

A045836 Half of theta series of b.c.c. lattice with respect to long edge.

Original entry on oeis.org

1, 2, 0, 0, 4, 4, 0, 0, 5, 4, 0, 0, 4, 8, 0, 0, 8, 6, 0, 0, 8, 4, 0, 0, 5, 12, 0, 0, 12, 8, 0, 0, 8, 8, 0, 0, 4, 12, 0, 0, 16, 8, 0, 0, 12, 8, 0, 0, 9, 14, 0, 0, 12, 16, 0, 0, 8, 4, 0, 0, 12, 16, 0, 0, 16, 16, 0, 0, 16, 8, 0, 0, 8, 20, 0, 0, 16, 8, 0, 0, 17
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The body-centered cubic (b.c.c. also known as D3*) lattice is the set of all triples [a, b, c] where the entries are all integers or all one half an odd integer. A long edge is centered at a triple with two integer entries and the remaining entry is one half an odd integer. - Michael Somos, May 31 2012

Examples

			q + 2*q^2 + 4*q^5 + 4*q^6 + 5*q^9 + 4*q^10 + 4*q^13 + 8*q^14 + 8*q^17 + ...

Links

Crossrefs

Cf. A004025, A045828, A045834.

Programs

  • Mathematica
    a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[QPochhammer[x^2+A]^5 * (QPochhammer[x^8+A]^4 / (QPochhammer[x+A]^2*QPochhammer[x^4+A]^4)), {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 05 2015, adapted from PARI *)
  • PARI
    {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A)^4 / (eta(x + A)^2 * eta(x^4 + A)^4), n))} /* Michael Somos, May 31 2012 */

Formula

From Michael Somos, May 31 2012: (Start)
Expansion of x * phi(x) * psi(x^4)^2 = x * psi(-x^2)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2)^5 * eta(q^8)^4 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Euler transform of period 8 sequence [ 2, -3, 2, 1, 2, -3, 2, -3, ...].
a(4*n) = a(4*n + 3) = 0. a(n) = A004025(n) / 2. a(4*n + 1) = A045834(n). a(4*n + 2) = 2 * A045828(n). (End)

A117728 A117726(n)/2.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 2, 4, 5, 4, 6, 4, 4, 8, 4, 4, 8, 6, 6, 8, 8, 4, 6, 8, 5, 12, 8, 4, 12, 8, 6, 8, 8, 8, 12, 10, 4, 12, 8, 8, 16, 8, 6, 12, 12, 8, 10, 8, 9, 14, 12, 8, 12, 16, 8, 16, 8, 4, 18, 8, 12, 16, 10, 8, 16, 16, 6, 16, 16, 8, 14, 12, 8, 20, 14, 12, 16, 8, 10, 16, 17, 8, 18, 16, 8, 20, 12
Offset: 1

Views

Author

N. J. A. Sloane, Apr 14 2006

Keywords

Examples

			G.f. = x + 2*x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 4*x^6 + 2*x^7 + 4*x^8 + 5*x^9 + ...

Crossrefs

Cf. A005884, A045834.

Programs

  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( sum(k=1, sqrtint(4*n + 9)\2, x^(k^2 + k - 2) / (1 - x^(2*k - 1))^2, A) / sum(k=1, sqrtint(4*n + 1)\2 + 1, x^(k^2 - k), A), n))}; /* Michael Somos, Jul 05 2015 */

Formula

a(4*n) = 2 * a(n). a(4*n + 1) = A045834(n). a(4*n + 2) = A005884(n). - Michael Somos, Jul 05 2015
G.f.: (Sum_{k>0} x^(k^2 + k - 1) / (1 - x^(2*k - 1))^2) / (Sum_{k>0} x^(k*(k - 1))). - Michael Somos, Jul 05 2015

A319078 Expansion of phi(-q) * phi(q)^2 in powers of q where phi() is a Ramanujan theta function.

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
Offset: 0

Views

Author

Michael Somos, Sep 09 2018

Keywords

Comments

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

Examples

			G.f. = 1 + 2*x - 4*x^2 - 8*x^3 + 6*x^4 + 8*x^5 - 8*x^6 + 12*x^8 + ...

Links

Crossrefs

Cf. A004015, A004024, A005887, A008443, A045834, A083703, A080965, A132429, A212885, A213624.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(16), 3/2), 66); A[1] + 2*A[2] - 4*A[3] - 8*A[4];
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q]^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2]^2, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^4), n))};

Formula

Expansion of eta(q^2)^9 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Expansion of phi(q) * phi(-q^2)^2 = phi(-q^2)^4 / phi(-q) in powers of q.
Euler transform of period 4 sequence [2, -7, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(11/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A045834.
G.f. Product_{k>0} (1 - x^k)^3 * (1 + x^k)^5 / (1 + x^(2*k))^4.
a(n) = (-1)^n * A212885(n) = A083703(2*n) = A080965(2*n).
a(4*n) = a(n) * -A132429(n + 2) where A132429 is a period 4 sequence.
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) = A005887(n). a(8*n + 5) = 2 * A004024(n). a(8*n + 6) = -8 * A213624(n). a(8*n + 7) = 0.

A329290 Number of ordered triples (i, j, k) of integers such that n = i^2 + 4*j^2 + 4*k^2.

Original entry on oeis.org

1, 2, 0, 0, 6, 8, 0, 0, 12, 10, 0, 0, 8, 8, 0, 0, 6, 16, 0, 0, 24, 16, 0, 0, 24, 10, 0, 0, 0, 24, 0, 0, 12, 16, 0, 0, 30, 8, 0, 0, 24, 32, 0, 0, 24, 24, 0, 0, 8, 18, 0, 0, 24, 24, 0, 0, 48, 16, 0, 0, 0, 24, 0, 0, 6, 32, 0, 0, 48, 32, 0, 0, 36, 16, 0, 0, 24, 32
Offset: 0

Views

Author

Michael Somos, Nov 17 2019

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(5/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A169783.

Examples

			G.f. = 1 + 2*x + 6*x^4 + 8*x^5 + 12*x^8 + 10*x^9 + 8*x^12 + 8*x^13 + ...

Links

Crossrefs

Cf. A005875, A045834, A169783.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(16), 3/2), 77); A[1] + 2*A[2];
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, x] EllipticTheta[3, 0, x^4]^2, {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A)^10 / (eta(x + A)^2 * eta(x^4 + A)^6 * eta(x^16 + A)^4), n))};

Formula

Euler transform of period 16 sequence [2, -3, 2, 3, 2, -3, 2, -7, 2, -3, 2, 3, 2, -3, 2, -3, ...].
Expansion of phi(x) * phi(x^4)^2 = phi(x^4)^3 + 2*x*phi(x^4)*psi(x^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.
G.f.: theta_3(q) * theta_3(q^4)^2, where theta_3() is the Jacobi theta function.
a(4*n) = A005875(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = a(4*n + 3) = 0.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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