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.

A077973 Expansion of 1/(1+x-2*x^3).

Original entry on oeis.org

1, -1, 1, 1, -3, 5, -3, -3, 13, -19, 13, 13, -51, 77, -51, -51, 205, -307, 205, 205, -819, 1229, -819, -819, 3277, -4915, 3277, 3277, -13107, 19661, -13107, -13107, 52429, -78643, 52429, 52429, -209715, 314573, -209715, -209715, 838861, -1258291, 838861, 838861, -3355443, 5033165, -3355443
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002, Jun 17 2007

Keywords

Links

Crossrefs

Cf. A077950, A134142.

Programs

  • GAP
    a:=[1,-1,1];; for n in [4..50] do a[n]:=-a[n-1]+2*a[n-3]; od; a; # G. C. Greubel, Jun 24 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1+x-2*x^3) )); // G. C. Greubel, Jun 24 2019
  • Mathematica
    LinearRecurrence[{-1,0,2}, {1,-1,1}, 50] (* or *) CoefficientList[Series[ 1/(1+x-2x^3), {x,0,50}], x] (* Harvey P. Dale, Jun 09 2017 *)
  • PARI
    Vec(1/(1+x-2*x^3)+O(x^50)) \\ Charles R Greathouse IV, Sep 23 2012
  • Sage
    (1/(1+x-2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 24 2019

Formula

a(n) = -a(n-1) +2*a(n-3). - Paul Curtz, Apr 23 2008
a(n) = (-1)^n * A077950(n) = (1/5) * (2*A134142(n) + 1). - Ralf Stephan, Aug 18 2013
G.f.: G(0)/(2*(1-x^2)), where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
a(n) = (-1)^n * Sum_{k=1..floor((n+2)/2)} binomial(n+2-2*k, k-1)*(-2)^(k-1). - Taras Goy, Sep 18 2019

A078014 Expansion of (1-x)/(1-x+2*x^3).

Original entry on oeis.org

1, 0, 0, -2, -2, -2, 2, 6, 10, 6, -6, -26, -38, -26, 26, 102, 154, 102, -102, -410, -614, -410, 410, 1638, 2458, 1638, -1638, -6554, -9830, -6554, 6554, 26214, 39322, 26214, -26214, -104858, -157286, -104858, 104858, 419430, 629146, 419430, -419430, -1677722, -2516582, -1677722, 1677722
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Links

Crossrefs

Cf. A077950.

Programs

  • GAP
    a:=[1,0,0];; for n in [4..50] do a[n]:=a[n-1]-2*a[n-3]; od; a; # G. C. Greubel, Jun 29 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x+2*x^3) )); // G. C. Greubel, Jun 29 2019
  • Mathematica
    LinearRecurrence[{1,0,-2}, {1,0,0}, 50] (* or *) CoefficientList[Series[ (1-x)/(1-x+2*x^3), {x,0,50}], x] (* G. C. Greubel, Jun 29 2019 *)
  • PARI
    my(x='x+O('x^50)); Vec((1-x)/(1-x+2*x^3)) \\ G. C. Greubel, Jun 29 2019
  • Sage
    ((1-x)/(1-x+2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 29 2019

Formula

G.f.: (1-x)/(1-x+2*x^3).
G.f.: G(0)/(2*(1+x)), where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
a(n) = A077950(n) - A077950(n-1). - R. J. Mathar, Jul 21 2016

A161909 Inverse binomial transform of A181586.

Original entry on oeis.org

0, 0, 1, -3, 7, -13, 23, -41, 79, -161, 335, -689, 1391, -2769, 5487, -10897, 21743, -43537, 87279, -174865, 349935, -699665, 1398511, -2795793, 5590767, -11182353, 22367983, -44740881, 89485039, -178966801, 357920495, -715821329, 1431629551, -2863272209, 5726596847
Offset: 0

Views

Author

Paul Curtz, Jan 31 2011

Keywords

Comments

A181586(n) and successive differences:
0, 0, 1, 0, 1, 2,
0, 1, -1, 1, 1, 3,
1, -2, 2, 0, 2, 0,
-3, 4, -2, 2, -2, 6,
7, -6, 4, -4, 8, -4,
-13, 10, -8, 12, -12, 12.
First column is a(n).

Links

Formula

a(n+2)=( (2*a(n)+A077950(n)) signed)
a(n)-a(n-1) = (-1)^n*A000749(n+2).
G.f. -x^2 / ( (x-1)*(2*x+1)*(2*x^2+2*x+1) ). - R. J. Mathar, Feb 06 2011
Showing 1-3 of 3 results.

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

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