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 1-3 of 3 results.

A155600 a(n) = 9^n-2^n+1^n.

Original entry on oeis.org

1, 8, 78, 722, 6546, 59018, 531378, 4782842, 43046466, 387419978, 3486783378, 31381057562, 282429532386, 2541865820138, 22876792438578, 205891132061882, 1853020188786306, 16677181699535498, 150094635296736978
Offset: 0

Views

Author

Mohammad K. Azarian, Jan 25 2009

Keywords

Crossrefs

Programs

  • Mathematica
    Table[9^n - 2^n + 1, {n, 0, 25}] (* or *)
    LinearRecurrence[{12, -29, 18}, {1, 8, 78}, 26] (* Paolo Xausa, Jul 19 2024 *)
  • PARI
    a(n)=9^n-2^n+1 \\ Charles R Greathouse IV, Sep 24 2015

Formula

G.f.: 1/(1-9*x)-1/(1-2*x)+1/(1-x). E.g.f.: e^(9*x)-e^(2*x)+e^x.
a(n) = 11*a(n-1)-18*a(n-2)+8 with a(0)=1, a(1)=8 - Vincenzo Librandi, Jul 21 2010
a(n) = A191465(n)+1. - R. J. Mathar, Mar 10 2022

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

Keywords

Crossrefs

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

A248216 a(n) = 6^n - 2^n.

Original entry on oeis.org

0, 4, 32, 208, 1280, 7744, 46592, 279808, 1679360, 10077184, 60465152, 362795008, 2176778240, 13060685824, 78364147712, 470184951808, 2821109841920, 16926659313664, 101559956406272, 609359739486208, 3656158439014400, 21936950638280704
Offset: 0

Views

Author

Vincenzo Librandi, Oct 04 2014

Keywords

Crossrefs

Sequences of the form k^n - 2^n: A001047 (k=3), A020522 (k=4), A005057 (k=5), this sequence (k=6), A190540 (k=7), A248217 (k=8), A191465 (k=9), A060458 (k=10), A139740 (k=11).

Programs

  • Magma
    [6^n-2^n: n in [0..25]];
    
  • Mathematica
    Table[6^n - 2^n, {n, 0, 25}] (* or *) CoefficientList[Series[4x/((1-2x)(1-6x)), {x, 0, 30}], x]
    LinearRecurrence[{8,-12},{0,4},30] (* Harvey P. Dale, Dec 21 2019 *)
  • Sage
    [2^n*(3^n -1) for n in (0..25)] # G. C. Greubel, Feb 09 2021

Formula

G.f.: 4*x/((1-2*x)*(1-6*x)).
a(n) = 8*a(n-1) - 12*a(n-2).
a(n) = 2^n*(3^n - 1) = A000079(n) * A024023(n).
E.g.f.: exp(6*x) - exp(2*x) = 2*exp(4*x)*sinh(2*x). - G. C. Greubel, Feb 09 2021
a(n) = 4*A016129(n-1). - R. J. Mathar, Mar 10 2022
a(n) = A000400(n) - A000079(n). - Bernard Schott, Mar 27 2022
Showing 1-3 of 3 results.