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.

A077952 Expansion of 1/(1 - x + x^2 + 2*x^3).

Original entry on oeis.org

1, 1, 0, -3, -5, -2, 9, 21, 16, -23, -81, -90, 37, 289, 432, 69, -941, -1874, -1071, 2685, 7504, 6961, -5913, -27882, -35891, 3817, 95472, 163437, 60331, -294050, -681255, -507867, 761488, 2631865, 2886111, -1268730, -9418571, -13922063, -1966032, 30793173, 60603331, 33742222, -88447455
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Comments

Row sums of Riordan array (1, x*(1-x-2*x^2)). - Paul Barry, Mar 09 2006

Links

Programs

  • GAP
    a:=[1,1,0];; for n in [4..50] do a[n]:=a[n-1]-a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Aug 07 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x+x^2+2*x^3) )); // G. C. Greubel, Aug 07 2019
  • Maple
    seq(coeff(series(1/(1-x+x^2+2*x^3), x, n+1), x, n), n = 0 .. 50); # G. C. Greubel, Aug 07 2019
  • Mathematica
    LinearRecurrence[{1,-1,-2}, {1,1,0}, 50] (* or *) CoefficientList[Series[ 1/(1-x+x^2+2*x^3), {x,0,50}], x] (* G. C. Greubel, Aug 07 2019 *)
  • PARI
    Vec(1/(1-x+x^2+2*x^3)+O(x^50)) \\ Charles R Greathouse IV, Sep 27 2012
  • Sage
    (1/(1-x+x^2+2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Aug 07 2019

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..n} C(k,j-k)*C(k,n-j)*(-2)^(n-j). - Paul Barry, Mar 09 2006
a(n) = (-1)^n*A077975(n). - R. J. Mathar, Jul 31 2010

A078045 Expansion of (1-x)/(1+x+x^2-2*x^3).

Original entry on oeis.org

1, -2, 1, 3, -8, 7, 7, -30, 37, 7, -104, 171, -53, -326, 721, -501, -872, 2815, -2945, -1614, 10189, -14465, 1048, 33795, -63773, 32074, 99289, -258909, 223768, 233719, -975305, 1189122, 253621, -3393353, 5517976, -1617381, -10687301, 23340634, -15888095, -28827141, 91396504
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Links

Crossrefs

First differences of A077975.

Programs

  • Mathematica
    CoefficientList[Series[(1-x)/(1+x+x^2-2*x^3),{x,0,40}],x] (* or *) LinearRecurrence[{-1,-1,2},{1,-2,1},50] (* Harvey P. Dale, Jun 12 2016 *)

Formula

Recurrence: a(n) = 2a(n-3) - a(n-2) - a(n-1), starting 1,-2,1. - Ralf Stephan, Aug 18 2013

A103749 Expansion of x*(1+2*x)/(1+x+x^2-2*x^3).

Original entry on oeis.org

0, 1, 1, -2, 3, 1, -8, 13, -3, -26, 55, -35, -72, 217, -215, -146, 795, -1079, -8, 2677, -4827, 2134, 8047, -19835, 16056, 19873, -75599, 87838, 27507, -266543, 414712, -93155, -854643, 1777222, -1108889, -2377619, 7040952, -6881111, -4915079, 25878094, -34725237, -983015
Offset: 0

Views

Author

Roger L. Bagula, Mar 28 2005

Keywords

Comments

Insert n=3, p=2, q=-1 and r=1 in Kenyon's characteristic polynomial x^n-p*x^(n-1)+q*x+r=0 .

Links

Crossrefs

Cf. A099206.

Programs

  • Mathematica
    LinearRecurrence[{-1,-1,2},{0,1,1},50] (* Harvey P. Dale, Jul 23 2013 *)

Formula

a(n) = -a(n-1)-a(n-2)+2*a(n-3) = A077975(n-1)+2*A077975(n-2).

Extensions

Edited, replaced by signed variant by the Assoc. Eds. of the OEIS - Jul 31 2010

A331890 a(n) = -a(n-1) - a(n-2) + 2*a(n-3) with a(0)=3, a(1)=-1, a(2)=-1.

Original entry on oeis.org

3, -1, -1, 8, -9, -1, 26, -43, 15, 80, -181, 131, 210, -703, 755, 368, -2529, 3671, -406, -8323, 16071, -8560, -24157, 64859, -57822, -55351, 242891, -303184, -50409, 839375, -1395334, 455141, 2618943, -5864752, 4156091, 6946547, -22832142, 24197777
Offset: 0

Views

Author

Wojciech Florek, Jan 30 2020

Keywords

Comments

a(n) is the reflected sequence (cf. A074058) of the generalized tribonacci sequence b(n) with b(0) = 3 and b(n) = A186575(n-1) for n > 0.

Links

Crossrefs

Cf. A074058, A186575.

Programs

  • Magma
    a:=[3,-1,-1]; [n le 3 select a[n] else -Self(n-1)-Self(n-2)+2*Self(n-3):n in [1..30]]; // Marius A. Burtea, Feb 02 2020
  • Mathematica
    LinearRecurrence[{-1,-1,2},{3,-1,-1},38] (* Stefano Spezia, Jan 31 2020 *)

Formula

G.f.: (3 + 2*x + x^2)/(1 + x + x^2 - 2*x^3).
a(n) = 3*A077975(n)+2*A077975(n-1)+A077975(n-2). - R. J. Mathar, Feb 28 2020

Extensions

Definition clarified by N. J. A. Sloane, Apr 23 2020
Showing 1-4 of 4 results.

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

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