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.

A171557 a(n) = 3^n*A168053(n).

Original entry on oeis.org

1, 3, -9, -81, -243, -1215, -5103, -15309, -59049, -216513, -649539, -2302911, -7971615, -23914845, -81310473, -272629233, -817887699, -2711943423, -8910671247, -26732013741, -87169610025, -282429536481, -847288609443
Offset: 0

Views

Author

Paul Barry, Dec 11 2009

Keywords

Comments

Hankel transform of A171556.

Links

Crossrefs

Cf. A168053, A171556.

Programs

  • Mathematica
    CoefficientList[Series[(1 - 18 x^2 - 81 x^3)/((1 - 3 x)^2*(1 + 3 x + 9 x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 21 2017 *)
  • PARI
    Vec((1-18*x^2-81*x^3)/((1-3*x)^2*(1+3*x+9*x^2)) + O(x^30)) \\ Michel Marcus, Feb 10 2015

Formula

G.f.: (1-18*x^2-81*x^3)/((1-3*x)^2*(1+3*x+9*x^2)).

A168056 Expansion of (1+2*x^2+x^3)/((1-x)^2*(1+x+x^2)).

Original entry on oeis.org

1, 1, 3, 5, 5, 7, 9, 9, 11, 13, 13, 15, 17, 17, 19, 21, 21, 23, 25, 25, 27, 29, 29, 31, 33, 33, 35, 37, 37, 39, 41, 41, 43, 45, 45, 47, 49, 49, 51, 53, 53, 55, 57, 57, 59, 61, 61, 63, 65, 65, 67, 69, 69, 71, 73, 73, 75, 77, 77, 79, 81, 81, 83, 85, 85, 87, 89, 89, 91, 93, 93
Offset: 0

Views

Author

Paul Barry, Nov 17 2009

Keywords

Links

Crossrefs

Cf. A168053.

Programs

  • Magma
    I:=[1,1,3,5]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Jul 08 2016
  • Mathematica
    LinearRecurrence[{1, 0, 1, -1}, {1, 1, 3, 5}, 100] (* G. C. Greubel, Jul 07 2016 *)
CoefficientList[Series[(1 + 2 x^2 + x^3) / ((1 - x)^2 (1 + x + x^2)), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 08 2016 *)

Formula

G.f.: (1+2*x^2+x^3)/((1-x)^2*(1+x+x^2)).
a(n) = A168057(n)/2^n.
a(n) = (12*n+3+6*cos(2*n*Pi/3)-2*sqrt(3)*sin(2*n*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017

A168054 Expansion of (1-8x^2-24x^3)/((1-2x)^2*(1+2x+4x^2)).

Original entry on oeis.org

1, 2, -4, -24, -48, -160, -448, -896, -2304, -5632, -11264, -26624, -61440, -122880, -278528, -622592, -1245184, -2752512, -6029312, -12058624, -26214400, -56623104, -113246208, -243269632, -520093696, -1040187392, -2214592512
Offset: 0

Views

Author

Paul Barry, Nov 17 2009

Keywords

Comments

Hankel transform of A168055.

Links

Crossrefs

Cf. A168053, A168055.

Programs

  • Magma
    I:=[1,2,-4,-24]; [n le 4 select I[n] else 2*Self(n-1)+8*Self(n-3)-16*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jul 08 2016
  • Mathematica
    LinearRecurrence[{2, 0, 8, -16}, {1, 2, -4, -24}, 100] (* G. C. Greubel, Jul 07 2016 *)
CoefficientList[Series[(1 - 8 x^2 - 24 x^3) / ((1 - 2 x)^2 (1 + 2 x + 4 x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 08 2016 *)

Formula

a(n) = 2^n*A168053(n).
a(n) = 2*a(n-1) + 8*a(n-3) - 16*a(n-4) for n>3. - Vincenzo Librandi, Jul 08 2016

A168071 Expansion of (1-3*x^2-4*x^3)/((1-x)^2*(1+x+x^2)).

Original entry on oeis.org

1, 1, -2, -5, -5, -8, -11, -11, -14, -17, -17, -20, -23, -23, -26, -29, -29, -32, -35, -35, -38, -41, -41, -44, -47, -47, -50, -53, -53, -56, -59, -59, -62, -65, -65, -68, -71, -71, -74, -77, -77, -80, -83, -83, -86, -89, -89, -92, -95, -95, -98, -101, -101, -104, -107, -107, -110, -113, -113
Offset: 0

Views

Author

Paul Barry, Nov 18 2009

Keywords

Links

Crossrefs

Cf. A168053.

Programs

  • Mathematica
    LinearRecurrence[{1, 0, 1, -1}, {1, 1, -2, -5}, 50] (* G. C. Greubel, Jul 08 2016 *)
  • PARI
    Vec((1-3*x^2-4*x^3)/((1-x)^2*(1+x+x^2)) + O(x^70)) \\ Michel Marcus, Dec 03 2014

Formula

G.f.: (1-3*x^2-4*x^3)/((1-x)^2*(1+x+x^2)).
a(n) = A168072(n)/3^n.
From Wesley Ivan Hurt, Oct 05 2017: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 3.
a(n) = (45 - 48*n + 18*cos(2*(n-1)*Pi/3) - 9*cos(Pi*cos(2*(n-1)*Pi/3) + Pi*sin(2*(n-1)*Pi/3)/sqrt(3)) + 14*sqrt(3)*sin(2*(n-1)*Pi/3))/24. (End)

Extensions

Corrected by R. J. Mathar, Dec 03 2014

A222657 a(n) = 2 * floor( (2*n + 1) / 3) + 1.

Original entry on oeis.org

1, 3, 3, 5, 7, 7, 9, 11, 11, 13, 15, 15, 17, 19, 19, 21, 23, 23, 25, 27, 27, 29, 31, 31, 33, 35, 35, 37, 39, 39, 41, 43, 43, 45, 47, 47, 49, 51, 51, 53, 55, 55, 57, 59, 59, 61, 63, 63, 65, 67, 67, 69, 71, 71, 73, 75, 75, 77, 79, 79, 81, 83, 83, 85, 87, 87, 89
Offset: 0

Views

Author

Michael Somos, May 29 2013

Keywords

Comments

Dimension of the space of weight 2n+4 cusp forms for Gamma_0(7).

Examples

			G.f. = 1 + 3*x + 3*x^2 + 5*x^3 + 7*x^4 + 7*x^5 + 9*x^6 + 11*x^7 + 11*x^8 + ...

Links

Crossrefs

Cf. A168053, A168056.

Programs

  • Mathematica
    Table[2Floor[(2n+1)/3]+1,{n,0,70}] (* or *) LinearRecurrence[{1,0,1,-1},{1,3,3,5},70] (* Harvey P. Dale, Sep 17 2024 *)
  • PARI
    {a(n) = (2*n + 1) \ 3 * 2 + 1};
  • Sage
    def a(n) : return( len( CuspForms( Gamma0( 7), 2*n + 4, prec=1). basis()));

Formula

G.f.: (1 + 2*x + x^3) / (1 - x - x^3 + x^4).
a(-n) = - A168056(n - 1).
a(n) = - A168053(n + 2).
a(n+3) = a(n) + 4.
a(n) = (12*n+9+4*sqrt(3)*sin(2*n*Pi/3))/9. - Wesley Ivan Hurt, Sep 29 2017
Showing 1-5 of 5 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.