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.

A083589 Expansion of 1/((1-4*x)*(1-x^4)).

Original entry on oeis.org

1, 4, 16, 64, 257, 1028, 4112, 16448, 65793, 263172, 1052688, 4210752, 16843009, 67372036, 269488144, 1077952576, 4311810305, 17247241220, 68988964880, 275955859520, 1103823438081, 4415293752324, 17661175009296, 70644700037184
Offset: 0

Views

Author

Paul Barry, May 02 2003

Keywords

Links

Crossrefs

Cf. A033139, A000975.

Programs

  • Mathematica
    CoefficientList[Series[1/((1-4x)(1-x^4)),{x,0,30}],x] (* or *) LinearRecurrence[ {4,0,0,1,-4},{1,4,16,64,257},31] (* Harvey P. Dale, Sep 13 2011 *)
  • PARI
    a(n)=(4^(n+4)+64)\255 \\ Charles R Greathouse IV, Jul 09 2013

Formula

a(0)=1, a(n) = 4*a(n-1) if n is not a multiple of 4, otherwise a(n) = 4*a(n-1) + 1. - Vincenzo Librandi, Mar 19 2011
a(n) = 4^(n+4)/255 -1/12 +(-1)^n/20 +(-1)^floor(n/2)*A010685(n)/34. - R. J. Mathar, Mar 19 2011
a(0)=1, a(1)=4, a(2)=16, a(3)=64, a(4)=257, a(n) = 4*a(n-1) + a(n-4) - 4*a(n-5). - Harvey P. Dale, Sep 13 2011
a(n) = floor(64*(2^(2*(n+1))+1)/255). - Tani Akinari, Jul 09 2013

A083590 Expansion of 1/((1-5*x)*(1-x^5)).

Original entry on oeis.org

1, 5, 25, 125, 625, 3126, 15630, 78150, 390750, 1953750, 9768751, 48843755, 244218775, 1221093875, 6105469375, 30527346876, 152636734380, 763183671900, 3815918359500, 19079591797500, 95397958987501, 476989794937505
Offset: 0

Views

Author

Paul Barry, May 02 2003

Keywords

Links

Crossrefs

Cf. A083589, A033139, A000975.

Programs

  • Magma
    [Floor(625*(5^(n+1)+1)/3124): n in [0..40]]; // G. C. Greubel, Oct 10 2017
  • Mathematica
    CoefficientList[Series[1/((1-5x)(1-x^5)),{x,0,40}],x] (* Vincenzo Librandi, Apr 04 2012 *)
LinearRecurrence[{5,0,0,0,1,-5},{1,5,25,125,625,3126},30] (* Harvey P. Dale, Jan 21 2023 *)
  • PARI
    Vec(1/(1-5*x)/(1-x^5)+O(x^99)) \\ Charles R Greathouse IV, Apr 04 2012

Formula

a(n) = floor(625*(5^(n+1)+1)/3124). - Tani Akinari, Jul 09 2013

A368344 a(n) = Sum_{k=0..n} 3^(n-k) * floor(k/3).

Original entry on oeis.org

0, 0, 0, 1, 4, 13, 41, 125, 377, 1134, 3405, 10218, 30658, 91978, 275938, 827819, 2483462, 7450391, 22351179, 67053543, 201160635, 603481912, 1810445743, 5431337236, 16294011716, 48882035156, 146646105476, 439938316437, 1319814949320, 3959444847969, 11878334543917
Offset: 0

Views

Author

Seiichi Manyama, Dec 22 2023

Keywords

Links

Crossrefs

Partial sums of A033139.
Column k=3 of A368343.
Cf. A097137.

Programs

  • PARI
    a(n, m=3, k=3) = (k^(n+1)\(k^m-1)-(n+1)\m)/(k-1);

Formula

a(n) = a(n-3) + (3^(n-2) - 1)/2.
a(n) = 1/2 * Sum_{k=0..n} floor(3^k/13) = Sum_{k=0..n} floor(3^k/26).
a(n) = 4*a(n-1) - 3*a(n-2) + a(n-3) - 4*a(n-4) + 3*a(n-5).
G.f.: x^3/((1-x) * (1-3*x) * (1-x^3)).
a(n) = (floor(3^(n+1)/26) - floor((n+1)/3))/2.

A083591 Inverse binomial transform of A083589.

Original entry on oeis.org

1, 3, 9, 27, 82, 242, 736, 2188, 6600, 19736, 59328, 177744, 533728, 1600160, 4802560, 14403520, 43218816, 129640064, 388952832, 1166793216, 3500510464, 10501269248, 31504332544, 94511948032, 283537942272, 850609632512
Offset: 0

Views

Author

Paul Barry, May 02 2003

Keywords

Links

Crossrefs

Cf. A033139.

Programs

  • Mathematica
    LinearRecurrence[{-1,6,14,12},{1,3,9,27,82},30] (* Harvey P. Dale, Nov 04 2024 *)

Formula

O.g.f.: -(1+x)^4/[(2*x+1)(2*x^2+2*x+1)(-1+3*x)]. - R. J. Mathar, Apr 02 2008
Showing 1-4 of 4 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.