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 1-2 of 2 results.

A016133 Expansion of 1/((1-2*x)*(1-9*x)).

Original entry on oeis.org

1, 11, 103, 935, 8431, 75911, 683263, 6149495, 55345711, 498111911, 4483008223, 40347076055, 363123688591, 3268113205511, 29413018865983, 264717169826615, 2382454528505071, 21442090756676711, 192978816810352543, 1736809351293697175, 15631284161644323151
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016131, A016136.
Cf. A016204 (partial sums); A191465 (this sequence times 7).

Programs

  • Magma
    [+9^(n+1)/7 -2^(n+1)/7 : n in [0..20]]; // Vincenzo Librandi, Aug 14 2011
  • Mathematica
    CoefficientList[Series[1/((1-2x)(1-9x)),{x,0,30}],x] (* or *) LinearRecurrence[ {11,-18},{1,11},30] (* Harvey P. Dale, Apr 19 2020 *)
  • PARI
    Vec(1/((1-2*x)*(1-9*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012
  • Sage
    [lucas_number1(n,11,18) for n in range(1, 20)] # Zerinvary Lajos, Apr 27 2009

Formula

a(n) = 11*a(n-1) - 18*a(n-2).
a(n) = a(n) = 9*a(n-1) + 2^n. - Paul Curtz, Feb 14 2008
E.g.f.: exp(2*x)*(9*exp(7*x) - 2)/7. - Stefano Spezia, Jul 30 2022

A016633 Expansion of g.f. 1/((1-2*x)*(1-11*x)*(1-12*x)).

Original entry on oeis.org

1, 25, 447, 6989, 101759, 1417941, 19180519, 253983853, 3309800367, 42599540357, 542895780791, 6863463633117, 86197420501375, 1076563471968373, 13382900349107463, 165700329729679181, 2044564737700501583, 25152545442794015589, 308625999807796411735, 3778261997130507936445
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A006516, A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009
Cf. A016135, A016136.

Programs

  • Magma
    [(648*12^n +2^(n+1)-5*11^(n+2))/45 : n in [0..20]]; // Vincenzo Librandi, Oct 09 2011
  • Mathematica
    CoefficientList[Series[1/((1 - 2 x) (1 - 11 x) (1 - 12 x)), {x, 0, 15}], x] (* Michael De Vlieger, Jan 31 2018 *)
  • PARI
    Vec(1/((1-2*x)*(1-11*x)*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012
  • Sage
    [(12^n - 2^n)/10-(11^n - 2^n)/9 for n in range(2,18)] # Zerinvary Lajos, Jun 05 2009

Formula

From Vincenzo Librandi, Oct 09 2011: (Start)
a(n) = (648*12^n + 2^(n+1) - 5*11^(n+2))/45.
a(n) = 23*a(n-1) - 132*a(n-2) + 2^n.
a(n) = 25*a(n-1) - 178*a(n-2) + 264*a(n-3), n >= 3. (End)
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(2*x)*(648*exp(10*x) - 605*exp(9*x) + 2)/45.
a(n) = A016136(n+1) - A016135(n+1). (End)
Showing 1-2 of 2 results.

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

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