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.

A054881 Number of walks of length n along the edges of an octahedron starting and ending at a vertex and also ( with a(0)=0 ) between two opposite vertices.

Original entry on oeis.org

1, 0, 4, 8, 48, 160, 704, 2688, 11008, 43520, 175104, 698368, 2797568, 11182080, 44744704, 178946048, 715849728, 2863267840, 11453333504, 45812809728, 183252287488, 733007052800, 2932032405504, 11728121233408
Offset: 0

Views

Author

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Keywords

Links

Crossrefs

Cf. A001045, A003683, A054882, A054883, A054884, A054885, A246036.

Programs

  • Magma
    [(4^n+(-1)^n*2^(n+1)+3*0^n)/6: n in [0..30]]; // Vincenzo Librandi, Apr 23 2015
  • Mathematica
    CoefficientList[Series[(1-2*x-4*x^2)/((1+2x)*(1-4x)), {x,0,40}], x] (* L. Edson Jeffery, Apr 22 2015 *)
LinearRecurrence[{2,8}, {1,0,4}, 41] (* G. C. Greubel, Feb 06 2023 *)
  • SageMath
    [(4^n + (-1)^n*2^(n+1) + 3*0^n)/6 for n in range(31)] # G. C. Greubel, Feb 06 2023

Formula

a(n) = 4*A003683(n-1) + 0^n/2, n >= 0.
a(n) = (4^n + (-1)^n*2^(n+1) + 3*0^n)/6.
G.f.: (1/6)*(3 + 2/(1+2*x) + 1/(1-4*x)).
From L. Edson Jeffery, Apr 22 2015: (Start)
G.f.: (1-2*x-4*x^2)/((1+2*x)*(1-4*x)).
a(n) = 8*A246036(n-3) + 0^n/2, n >= 0. (End)
a(n) = 2^n*A001045(n-1) + (1/2)*[n=0] = 2^n*(2^(n-1) + (-1)^n)/3 + (1/2)*[n=0], n >= 0. - Ralf Steiner, Aug 27 2020, edited by M. F. Hasler, Sep 11 2020
E.g.f.: (1/6)*(exp(4*x) + 2*exp(-2*x) + 3). - G. C. Greubel, Feb 06 2023

A054882 Closed walks of length n along the edges of a dodecahedron based at a vertex.

Original entry on oeis.org

1, 0, 3, 0, 15, 6, 87, 84, 567, 882, 4095, 8448, 32079, 78078, 265863, 710892, 2282631, 6430794, 20009391, 58008216, 177478623, 522598230, 1584540279, 4705481220, 14198074455, 42357719586, 127472924127, 381253030704
Offset: 0

Views

Author

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Keywords

Links

Crossrefs

Cf. A054881, A054883, A054884, A054885.

Programs

  • Magma
    [Ceiling((5+3^n+(-1)^n*2^(n+2)+3*(1+(-1)^n)*Sqrt(5)^n)/20): n in [0..30]]; // Vincenzo Librandi, Aug 24 2011
  • Mathematica
    LinearRecurrence[{2,10,-16,-25,30}, {1,0,3,0,15,6}, 41] (* G. C. Greubel, Feb 07 2023 *)
  • SageMath
    def A054882(n): return (5+3^n+4*(-2)^n+3*(1+(-1)^n)*5^(n/2)+4*0^n)/20
[A054882(n) for n in range(41)] # G. C. Greubel, Feb 07 2023

Formula

G.f.: (1/20)*(4 + 5/(1-x) + 1/(1-3*x) + 4/(1+2*x) + 6/(1-5*x^2)).
G.f.: (1 - 2*x - 7*x^2 + 10*x^3 + 10*x^4 - 6*x^5)/((1-x)*(1+2*x)*(1-3*x)*(1-5*x^2)).
a(n) = (5 + 3^n + (-1)^n*2^(n+2) + 3*(1+(-1)^n)*sqrt(5)^n + 4*0^n)/20.
E.g.f.: (1/20)*(4 + 4*exp(-2*x) + 5*exp(x) + exp(3*x) + 6*cosh(sqrt(5)*x)). - G. C. Greubel, Feb 07 2023

A054884 Number of closed walks of length n along the edges of an icosahedron based at a vertex.

Original entry on oeis.org

1, 0, 5, 10, 65, 260, 1365, 6510, 32865, 162760, 815365, 4069010, 20352865, 101725260, 508665365, 2543131510, 12715852865, 63578287760, 317892415365, 1589457194010, 7947290852865, 39736429850260
Offset: 0

Views

Author

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Keywords

Links

Crossrefs

Cf. A030517, A030518, A054881, A054882, A054883, A054885.

Programs

  • Magma
    [Floor((5^n+(-1)^n*5+3*(1+(-1)^n)*Sqrt(5)^n)/12): n in [0..30]]; // Vincenzo Librandi, Aug 24 2011
  • Mathematica
    LinearRecurrence[{4,10,-20,-25},{1,0,5,10},30] (* Harvey P. Dale, May 02 2022 *)
  • PARI
    a(n) = if(n%2, 5^n-5, 5^n+5+6*5^(n/2))/12; \\ François Marques, Jul 11 2021
  • SageMath
    def A054884(n): return (5^n + 5*(-1)^n + 3*(1 + (-1)^n)*5^(n/2))/12
[A054884(n) for n in range(41)] # G. C. Greubel, Feb 07 2023

Formula

G.f.: (1/12)*(1/(1-5*t) + 5/(1+t) + 6/(1-5*t^2)).
a(n) = (5^n + (-1)^n*5 + 3*(1 + (-1)^n)*sqrt(5)^n)/12.
a(n+1) = 5 * A030517(n) for n > 0.
a(n) = 4*a(n-1) + 10*a(n-2) - 20*a(n-3) - 25*a(n-4). - François Marques, Jul 10 2021
E.g.f.: (1/12)*(5*exp(-x) + exp(5*x) + 6*cosh(sqrt(5)*x)). - G. C. Greubel, Feb 07 2023

A054885 Number of walks of length n along the edges of an icosahedron between two opposite vertices.

Original entry on oeis.org

0, 0, 0, 10, 40, 260, 1240, 6510, 32240, 162760, 812240, 4069010, 20337240, 101725260, 508587240, 2543131510, 12715462240, 63578287760, 317890462240, 1589457194010, 7947281087240, 39736429850260
Offset: 0

Views

Author

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Keywords

Links

Crossrefs

Cf. A030517, A030518, A054881, A054882, A054883, A054884.

Programs

  • Magma
    [Floor((5^n+(-1)^n*5-3*(1+(-1)^n)*Sqrt(5)^n)/12): n in [0..30]]; // Vincenzo Librandi, Aug 24 2011
  • Mathematica
    LinearRecurrence[{4,10,-20,-25}, {0,0,0,10}, 41] (* G. C. Greubel, Feb 07 2023 *)
  • PARI
    a(n) = if(n%2, 5^n-5, 5^n+5-6*5^(n/2))/12; \\ François Marques, Jul 11 2021
  • SageMath
    def A054885(n): return (5^n +5*(-1)^n -3*(1+(-1)^n)*5^(n/2))/12
[A054885(n) for n in range(41)] # G. C. Greubel, Feb 07 2023

Formula

G.f.: (1/12)*(1/(1-5*t) + 5/(1+t) - 6/(1-5*t^2)).
a(n) = (5^n + 5*(-1)^n - 3*(1 + (-1)^n)*sqrt(5)^n)/12.
a(n+1) = 5 * A030518(n) for n > 0.
a(n) = 4*a(n-1) + 10*a(n-2) - 20*a(n-3) - 25*a(n-4). - François Marques, Jul 10 2021
E.g.f.: (1/12)*(5*exp(-x) + exp(5*x) - 6*cosh(sqrt(5)*x)). - G. C. Greubel, Feb 07 2023

A375309 Number of walks of length n along the edges of a dodecahedron graph between two vertices at distance two.

Original entry on oeis.org

0, 0, 1, 1, 7, 11, 51, 105, 399, 967, 3299, 8789, 28271, 79443, 247507, 716353, 2193583, 6452639, 19575075, 58095597, 175350735, 522947755, 1574075603, 4706879321, 14146450127, 42363311991, 127217598691, 381275400325, 1144458922159
Offset: 0

Views

Author

Miquel A. Fiol, Aug 11 2024

Keywords

Links

Crossrefs

Cf. A054883.

Programs

  • Mathematica
    LinearRecurrence[{2, 10, -16, -25, 30}, {0, 0, 1, 1, 7, 11}, 30] (* Amiram Eldar, Aug 13 2024 *)

Formula

For n>=6, a(n) = 2*a(n-1) + 10*a(n-2) - 16*a(n-3) - 25*a(n-4) + 30*a(n-5).
From Stefano Spezia, Aug 13 2024: (Start)
G.f.: x^2*(1 - x - 5*x^2 + 3*x^3)/((1 - x)*(1 + 2*x)*(1 - 3*x)*(1 - 5*x^2)).
a(n) = (3*5^(n/2)*(1 + (-1)^n) + 3^(1+n) + (-1)^n*2^(1+n) - 5)/60 for n > 0. (End)
Showing 1-5 of 5 results.

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