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 1-5 of 5 results.

A014455 Theta series of quadratic form with Gram matrix [ 1, 0, 0; 0, 1, 0; 0, 0, 2 ]. Number of integer solutions to x^2 + y^2 + 2*z^2 = n.

Original entry on oeis.org

1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 24, 8, 24, 24, 0, 16, 12, 16, 30, 24, 24, 16, 24, 16, 8, 28, 24, 32, 48, 8, 0, 32, 6, 32, 48, 16, 36, 40, 24, 16, 24, 16, 48, 40, 24, 40, 0, 32, 24, 36, 30, 16, 72, 24, 32, 48, 0, 32, 72, 24, 48, 40, 0, 48, 12, 16, 48, 56, 48, 32, 48, 16, 30, 64
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This is the tetragonal P lattice (the classical holotype) of dimension 3.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 4*q + 6*q^2 + 8*q^3 + 12*q^4 + 8*q^5 + 8*q^6 + 16*q^7 + 6*q^8 + 12*q^9 + ...

Links

Crossrefs

Cf. A004015, A005875, A005877, A008443, A033763, A045826, A045828, A156384, A213024, A246631.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(8), 3/2), 40); A[1] + 4*A[2] + 6*A[3] + 8*A[4]; /* Michael Somos, Aug 31 2014 */
  • Mathematica
    r[n_, z_] := Reduce[x^2 + y^2 + 2*z^2 == n, {x, y}, Integers]; a[n_] := Module[{rn0, rnz, k0, k}, rn0 = r[n, 0]; k0 = If[rn0 === False, 0, If[Head[rn0] === And, 1, Length[rn0]]]; For[k = 0; z = 1, z <= Ceiling[Sqrt[n/2]], z++, rnz = r[n, z]; If[rnz =!= False, k = If[Head[rnz] === And, k+1, k + Length[rnz]]]]; k0 + 2*k]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 07 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2 EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Aug 31 2014 *)
QP = QPochhammer; s = QP[q^2]^8*(QP[q^4]/(QP[q]^4*QP[q^8]^2)) + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, after Michael Somos *)
  • PARI
    {a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0, 0; 0, 1, 0; 0, 0, 2], n)[n])}; /* Michael Somos, Jul 05 2005 */
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^4 + A) / (eta(x + A)^4 * eta(x^8 + A)^2), n))}; /* Michael Somos, Jul 05 2005 */

Formula

Expansion of phi(q)^2 * phi(q^2) = psi(q)^4 / psi(q^4) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Apr 07 2012
Expansion of eta(q^2)^8 * eta(q^4) / (eta(q)^4 * eta(q^8)^2) in powers of q. - Michael Somos, Jul 05 2005
Euler transform of period 8 sequence [4, -4, 4, -5, 4, -4, 4, -3, ...]. - Michael Somos, Jul 07 2005
G.f.: theta_3(q)^2 * theta_3(q^2) = Product_{k>0} (1 - x^(2*k))^8 * (1 - x^(4*k)) / ((1 - x^k)^4 * (1 - x^(8*k))^2).
There is a classical formula (essentially due to Gauss): Write (uniquely) -2n=D(2^vf)^2, with D<0 fundamental discriminant, f odd, v>=-1. Then a(n)=12L((D/.),0)(1-(D/2))\sum_{d\mid f}\mu(d)(D/d)sigma(f/d) (the formula for A005875), except that the factor (1-(D/2)) has to be replaced by 1/3 if v=-1 and by 1 if v=0 (and kept if v>=1). Here mu() is the Moebius function, (D/2) and (D/d) are Kronecker-Legendre symbols, sigma() is the sum of divisors function, L((D/.),0)=h(D)/(w(D)/2) is the value at 0 of the L() function of the quadratic character (D/.), equal to the class number h(D) divided by 2 or 3 in the special cases D=-4 and -3. - Henri Cohen (Henri.Cohen(AT)math.u-bordeaux1.fr), May 12 2010
a(2*n) = a(8*n) = A005875(n). a(2*n + 1) = A005877(n) = 4 * A045828(n). a(4*n) = A004015(n). a(4*n + 2) = 2 * A045826(n). a(8*n + 4) = 12 * A045828(n). a(8*n + 7) = 16 * A033763(n). a(16*n + 6) = 8 * A008443(n). a(16*n + 14) = 0. - Michael Somos, Apr 07 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246631.

A004013 Theta series of body-centered cubic (b.c.c.) lattice.

Original entry on oeis.org

1, 0, 0, 8, 6, 0, 0, 0, 12, 0, 0, 24, 8, 0, 0, 0, 6, 0, 0, 24, 24, 0, 0, 0, 24, 0, 0, 32, 0, 0, 0, 0, 12, 0, 0, 48, 30, 0, 0, 0, 24, 0, 0, 24, 24, 0, 0, 0, 8, 0, 0, 48, 24, 0, 0, 0, 48, 0, 0, 72, 0, 0, 0, 0, 6, 0, 0, 24, 48, 0, 0, 0, 36, 0, 0, 56, 24, 0, 0, 0, 24, 0, 0, 72, 48, 0, 0, 0, 24, 0, 0
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).
The bcc lattice is also called the coweight lattice A^*3 (or D^*_3), dual to the root lattice A_3 (or D_3, or the fcc lattice), or the permutohedral lattice A^*_3. - _Andrey Zabolotskiy, Mar 08 2020

Examples

			G.f. = 1 + 8*x^3 + 6*x^4 + 12*x^8 + 24*x^11 + 8*x^12 + 6*x^16 + 24*x^19 + 24*x^20 + ...
G.f. = 1 + 8*q^(3/2) + 6*q^2 + 12*q^4 + 24*q^(11/2) + 8*q^6 + 6*q^8 + 24*q^(19/2) + 24*q^10 + 24*q^12 + 32*q^(27/2) + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 116.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A004015, A005875, A008443, A045826, A045828, A045834, A213056.
Indices of nonzero terms are A004014.
Cf. A023916-A023936.
Cf. A004011, A008422, A008425, A008423, A008427, A008424, A008426.

Programs

  • Magma
    Basis( ModularForms( Gamma0(8), 3/2), 90) [1]; /* Michael Somos, Sep 04 2014 */
  • Maple
    M:=100; M1:=M*(M+1)/2; ph:=series(add(q^(k^2),k=-M..M),q,M1): ps:=series(add(q^(k*(k+1)/2),k=0..M),q,M1): t1:=series(subs(q=q^2, ph)^3, q,M1): t2:=series((2*sqrt(q))^3*subs(q=q^4, ps)^3,q,M1): t3:=seriestolist(series(subs(q=q^2,t1+t2),q,M1)): for n from 0 to nops(t3)-1 do lprint(n,t3[n+1]); od:
  • Mathematica
    m = 13; m1 = m*((m + 1)/2); ph[q_] = Series[ Sum[ q^k^2, {k, -m, m}], {q, 0, m1}]; ps[q_] = Series[ Sum[ q^(k*((k + 1)/2)), {k, 0, m}], {q, 0, m1}]; t1[q_] = Normal[ Series[ ph[q^2]^3, {q, 0, m1}]]; t2[q_] = Normal[ Series[ (2*Sqrt[q])^3*ps[q^4]^3, {q, 0, m1}]]; CoefficientList[ Series[ t1[q^2] + t2[q^2], {q, 0, m1}], q] (* Jean-François Alcover, Dec 20 2011, translated from Maple *)
(* From version 6 on *) terms=91; f[q_] = LatticeData["BodyCenteredCubic", "ThetaSeriesFunction"][-I Log[q]/Pi]; CoefficientList[Simplify[f[q] + O[q]^terms, q>0], q][[1 ;; terms]] (* Jean-François Alcover, May 15 2013, updated Jul 08 2017 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^4]^3 + EllipticTheta[ 2, 0, x^4]^3, {x, 0, n}]; (* Michael Somos, May 24 2013 *)
  • PARI
    {a(n) = if( n<0, 0, if( n%4==0, n/=4; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x * O(x^n))^3, n), n%8==3, n\=8; 8*polcoeff( sum(k=0, (sqrtint(8*n+1) - 1)\2, x^((k^2 + k)/2), x * O(x^n))^3, n)))}; /* Michael Somos, Oct 25 2006 */
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A)^5 / eta(x^4 + A)^2 / eta(x^16 + A)^2)^3 + (2 * x * eta(x^16 + A)^2 / eta(x^8 + A))^3, n))}; /* Michael Somos, May 17 2008 */

Formula

subs(q=q^2, ph)^3+(2*sqrt(q))^3*subs(q=q^4, ps)^3, where ps = A010054 = Sum_{k=0..infinity} q^(k*(k+1)/2), ph = A000122 = Sum_{k=-infinity, infinity} q^(k^2).
Expansion of phi(q^4)^3 + 8 * q^3 * psi(q^8)^3 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Oct 25 2006
a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = 0. a(4*n) = A005875(n).
Expansion of theta_3(q)^3 + theta_2(q)^3 in powers of q^(1/4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A004015.
a(8*n) = A004015(n). a(8*n + 3) = 8 * A008443(n). a(8*n + 4) = 2 * A045826(n). - Michael Somos, Jul 19 2015
a(12*n + 4) = 6 * A213056(n). a(16*n + 4) = 6 * A045834(n). a(16*n + 8) = 12 * A045828(n).

A213384 Expansion of phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 12, -8, 6, -24, 24, 0, 12, -30, 24, -24, 8, -24, 48, 0, 6, -48, 36, -24, 24, -48, 24, 0, 24, -30, 72, -32, 0, -72, 48, 0, 12, -48, 48, -48, 30, -24, 72, 0, 24, -96, 48, -24, 24, -72, 48, 0, 8, -54, 84, -48, 24, -72, 96, 0, 48, -48, 24, -72, 0, -72, 96
Offset: 0

Views

Author

Michael Somos, Jun 10 2012

Keywords

Comments

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

Examples

			G.f. = 1 - 6*q + 12*q^2 - 8*q^3 + 6*q^4 - 24*q^5 + 24*q^6 + 12*q^8 - 30*q^9 + ...

Links

Crossrefs

Cf. A004015, A005875, A008443, A045826, A045834.

Programs

  • Julia
    # JacobiTheta4 is defined in A002448.
A213384List(len) = JacobiTheta4(len, 3)
A213384List(63) |> println # Peter Luschny, Mar 12 2018
  • Magma
    A := Basis( ModularForms( Gamma0(16), 3/2), 63); A[1] - 6*A[2] + 12*A[3] - 8*A[4]; /* Michael Somos, May 21 2015 */
  • Mathematica
    a[ n_] := (-1)^n SquaresR[ 3, n]; (* Michael Somos, May 21 2015 *)
a[ n_] := (-1)^n Length @ FindInstance[ n == x^2 + y^2 + z^2, {x, y, z}, Integers, 10^9]; (* Michael Somos, May 21 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^3, {q, 0, n}]; (* Michael Somos, May 21 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 / QPochhammer[ q^2])^3, {q, 0, n}]; (* Michael Somos, May 21 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 / eta(x^2 + A))^3, n))};
  • PARI
    {a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), 2 * (-x)^k^2, 1 + x * O(x^n))^3, n))}; /* Michael Somos, May 21 2015 */
  • PARI
    {a(n) = my(G); if( n<0, 0, G = [ 1, 0, 0; 0, 1, 0; 0, 0, 1]; (-1)^n * polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, May 21 2015 */

Formula

Expansion of (eta(q)^2 / eta(q^2))^3 in powers of q.
Euler transform of period 2 sequence [ -6, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(15/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008443.
G.f.: (Sum_{k in Z} (-1)^k * x^k^2)^3.
a(n) = (-1)^n * A005875(n). a(2*n) = A004015(n). a(2*n + 1) = -2 * A045826(n). a(4*n) = A005875(n). a(4*n + 1) = -6 * A045834(n). a(4*n + 2) = 12 * A045828(n). a(8*n + 3) = -8 * A008443(n). a(8*n + 7) = 0.

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 1-5 of 5 results.

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