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 1-4 of 4 results.

A072835 Exponents occurring in expansion of F_9(q^2).

Original entry on oeis.org

0, 8, 14, 18, 20, 26, 32, 36, 38, 44, 50, 54, 56, 62, 68, 72, 74, 80, 86, 90, 92, 98, 104, 108, 110, 116, 122, 126, 128, 134, 140, 144, 146, 152, 158, 162, 164, 170, 176, 180, 182, 188, 194, 198, 200, 206, 212, 216, 218, 224, 230, 234, 236, 242, 248, 252, 254, 260, 266, 270, 272, 278
Offset: 0

Views

Author

N. J. A. Sloane, Jul 25 2002

Keywords

Comments

Twice (A242660 without 1). Also, norms of vectors of the A*8 lattice. - _Andrey Zabolotskiy, Nov 10 2021

Links

Crossrefs

Cf. A072839, A242660.
Cf. A004014, A047222, A072833, A047328, A072834, A072836.
Cf. A056991.

Programs

  • Mathematica
    f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y];
F[9,q_]:= f[q^9, q^9]^8 - 16*q^9*f[q^9, q^27]^8 + 256*q^18*f[q^18, q^54]^8 + 18*q^8*QPochhammer[q^18]^12/QPochhammer[q^6]^4;
cfs = CoefficientList[Series[F[9, q], {q, 0, 500}], q];
Take[Pick[Range[Length[cfs]] - 1, Sign[Abs[cfs]], 1], 50] (* G. C. Greubel, Apr 16 2018 *)

Formula

G.f.: -2*x*(x^4-x^3-2*x^2-3*x-4) / (x^5-x^4-x+1). - Colin Barker, Jul 31 2013
a(n+4) = a(n) + 18 for n > 0. - Jerzy R Borysowicz, Sep 02 2023
a(n)/n ~ 9/2. - Jerzy R Borysowicz, Sep 03 2023
a(n) = 2 * A056991(n+1) for n>=1. - Alois P. Heinz, Sep 03 2023

Extensions

Terms a(22) onward added by G. C. Greubel, Apr 16 2018

A134719 Odd Padovan numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 5, 7, 9, 21, 37, 49, 65, 151, 265, 351, 465, 1081, 1897, 2513, 3329, 7739, 13581, 17991, 23833, 55405, 97229, 128801, 170625, 396655, 696081, 922111, 1221537, 2839729, 4983377, 6601569, 8745217, 20330163, 35676949, 47261895, 62608681
Offset: 1

Views

Author

Omar E. Pol, Nov 11 2007

Keywords

Crossrefs

Cf. A000931.
Cf. A047328 (Indices of the odd numbers in the Padovan sequence). - Francesco Daddi, Jul 31 2011

Programs

  • Mathematica
    Select[CoefficientList[Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, 100}], x], OddQ] (* T. D. Noe, Jul 31 2011 *)

Formula

G.f.: x*(x+1)*(x^10-2*x^9+3*x^8-4*x^7+4*x^6-2*x^5+6*x^4-x^2-1) / (x^12+x^8+7*x^4-1). - Colin Barker, Sep 17 2013

A023918 Theta series of A*_6 lattice.

Original entry on oeis.org

1, 0, 0, 14, 0, 42, 70, 42, 0, 0, 210, 0, 294, 294, 210, 0, 0, 504, 0, 630, 882, 350, 0, 0, 1190, 0, 1470, 1148, 882, 0, 0, 1680, 0, 1708, 2520, 1050, 0, 0, 3150, 0, 3570, 2940, 1750, 0, 0, 3066, 0, 3864, 4774, 2100, 0, 0, 6174, 0, 5740, 5124, 3570, 0, 0, 6090
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Positions of nonzero entries seem to be A047328. - Andrey Zabolotskiy, Nov 10 2021

Examples

			1 + 14*x^3 + 42*x^5 + 70*x^6 + 42*x^7 + 210*x^10 + 294*x^12 + 294*x^13 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.

Links

Crossrefs

Cf. A008446.
Cf. theta series of lattices A*_0, A*_1, A*_2, A*_3, A*_4...: A000007, A000122, A004016, A004013, A023916, A023917, this sequence, A023919-A023936.
Cf. A047328.

Programs

  • Mathematica
    a[n_] := Module[{A, A7}, A = x*O[x]^n; A7 = QPochhammer[x^7 + A]; A = QPochhammer[x + A]; SeriesCoefficient[A^7 / A7 + 7 * x * (A * A7)^3 + 7 * x^2 * A7^7 / A, {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 05 2015, adapted from Michael Somos's PARI script *)
  • PARI
    {a(n) = local(A, A7); if( n<0, 0, A = x * O(x^n); A7 = eta(x^7 + A); A = eta(x + A); polcoeff( A^7 / A7 + 7 * x * (A * A7)^3 + 7 * x^2 * A7^7 / A, n))}; /* Michael Somos, Jan 29 2011 */

Formula

Expansion of f(-x)^7 / f(-x^7) + 7 * x * f(-x)^3 * f(-x^7)^3 + 7 * x^2 * f(-x^7)^7 / f(-x) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Jan 29 2011
a(7*n) = A008446(n). a(7*n + 1) = a(7*n + 2) = a(7*n + 4) = 0. - Michael Somos, Jan 29 2011

A028919 Congruent to 0, 6, 10, 12 (mod 14).

Original entry on oeis.org

0, 6, 10, 12, 14, 20, 24, 26, 28, 34, 38, 40, 42, 48, 52, 54, 56, 62, 66, 68, 70, 76, 80, 82, 84, 90, 94, 96, 98, 104, 108, 110, 112, 118, 122, 124, 126, 132, 136, 138, 140, 146, 150, 152, 154, 160, 164, 166, 168, 174, 178, 180, 182, 188, 192, 194, 196, 202, 206, 208, 210, 216, 220, 222, 224, 230, 234, 236
Offset: 1

Views

Author

Jan.Hagberg(AT)stat.su.se

Keywords

Links

Programs

  • Mathematica
    With[{c=14*Range[0,20]},Union[Flatten[{c,c+6,c+10,c+12}]]] (* Harvey P. Dale, Dec 25 2011 *)

Formula

a(n) = 2*A047328(n). G.f. 2*x^2*(3+2*x+x^2+x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

Extensions

Corrected by R. J. Mathar, Oct 08 2011
Showing 1-4 of 4 results.

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

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