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.

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A139790 a(n) = (5*2^(n+2) - 3*n*2^n - 2*(-1)^n) / 18.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 7, -7, -57, -199, -569, -1479, -3641, -8647, -20025, -45511, -101945, -225735, -495161, -1077703, -2330169, -5009863, -10718777, -22835655, -48467513, -102527431, -216239673, -454848967, -954437177, -1998352839, -4175662649, -8709239239
Offset: 0

Views

Author

Paul Curtz, May 21 2008

Keywords

Comments

Binomial transform of 1,1,0,1,-2,3,-8,13,-30,... (see A113954 and A103196). - R. J. Mathar, Feb 11 2010

Links

Crossrefs

Cf. A136298, A136299.

Programs

  • Magma
    [(5*2^(n+2)-3*n*2^n-2*(-1)^n) / 18: n in [0..35]]; // Vincenzo Librandi, Aug 09 2011
  • Mathematica
    LinearRecurrence[{3,0,-4},{1,2,3},40] (* Harvey P. Dale, May 27 2018 *)

Formula

a(n+1) - 2*a(n) = -A001045(n).
G.f.: (1 - x - 3*x^2)/((1+x)*(1-2*x)^2).
a(n) = 3*a(n-1) - 4*a(n-3).

Extensions

Definition replaced with closed formula by R. J. Mathar, Feb 11 2010

A348405 a(0) = 1, a(n) + a(n+1) = round(2^n/9), n >= 0.

Original entry on oeis.org

1, -1, 1, -1, 2, 0, 4, 3, 11, 17, 40, 74, 154, 301, 609, 1211, 2430, 4852, 9712, 19415, 38839, 77669, 155348, 310686, 621382, 1242753, 2485517, 4971023, 9942058, 19884104, 39768220, 79536427, 159072867, 318145721, 636291456, 1272582898, 2545165810
Offset: 0

Views

Author

Paul Curtz, Oct 17 2021

Keywords

Links

Crossrefs

Cf. A139797 (a(n) + a(n+1) = round(2^n/9) too, but a(0) = 0).
Cf. A001045, A104581, A113405, A131666.
Cf. A136298, A172481, A175656, A348407.

Programs

  • Mathematica
    CoefficientList[ Series[(x^4-x^3+2x-1)/((2*x^3-3*x^2+3*x-1)*(x+1)^2), {x, 0, 40}], x] (* Thomas Scheuerle, Oct 17 2021 *)
nxt[{n_,a_}]:={n+1,Round[(2^n)/9]-a}; NestList[nxt,{0,1},40][[All,2]] (* or *) LinearRecurrence[{1,2,-1,1,2},{1,-1,1,-1,2},40] (* Harvey P. Dale, Apr 28 2022 *)

Formula

a(n+1) = 2*a(n) - A104581(n+6).
a(n) + a(n+1) = A113405(n).
a(n) + a(n+3) = A001045(n).
a(n+2) = a(n) + A131666(n).
From Thomas Scheuerle, Oct 18 2021: (Start)
G.f.: (x^4-x^3+2x-1)/((2*x^3-3*x^2+3*x-1)*(x+1)^2).
A172481(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(2*n-k). With negative sign for ...*a(1+2*n-k) and ...*a(3+2*n-k) too.
A175656(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(2+2*n-k).
A136298(n+1) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(4+2*n-k).
A348407(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*(a(2+2*n-k) - 2*a(1+2*n-k) - a(2*n-k)).
(End)

Extensions

a(22)-a(36) from Thomas Scheuerle, Oct 17 2021
Showing 1-2 of 2 results.

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