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 1-5 of 5 results.

A077911 Expansion of 1/((1-x)*(1+x+2*x^2-x^3)).

Original entry on oeis.org

1, 0, -1, 3, 0, -6, 10, 3, -28, 33, 27, -120, 100, 168, -487, 252, 891, -1881, 352, 4302, -6886, -1365, 19440, -23595, -16649, 83280, -73576, -109632, 340065, -194376, -595385, 1324203, -327808, -2915982, 4895802, 608355, -13315940, 16995033, 10245203, -57551208, 54055836, 71291784
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Links

Crossrefs

Cf. A077978.

Programs

  • GAP
    a:=[1,0,-1,3];; for n in [4..50] do a[n]:=-a[n-2]+3*a[n-3]-a[n-4]; od; a; # G. C. Greubel, Jul 02 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1-x)*(1+x+2*x^2-x^3)) )); // G. C. Greubel, Jul 02 2019
  • Mathematica
    LinearRecurrence[{0,-1,3,-1}, {1,0,-1,3}, 50] (* or *) CoefficientList[ Series[1/((1-x)*(1+x+2*x^2-x^3)), {x,0,50}], x] (* G. C. Greubel, Jul 02 2019 *)
  • PARI
    Vec(1/((1-x)*(1+x+2*x^2-x^3))+O(x^50)) \\ Charles R Greathouse IV, Sep 27 2012
  • Sage
    (1/((1-x)*(1+x+2*x^2-x^3))).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jul 02 2019

A077955 Expansion of 1/(1-x+2*x^2+x^3).

Original entry on oeis.org

1, 1, -1, -4, -3, 6, 16, 7, -31, -61, -6, 147, 220, -68, -655, -739, 639, 2772, 2233, -3950, -11188, -5521, 20805, 43035, 6946, -99929, -156856, 36056, 449697, 534441, -401009, -1919588, -1652011, 2588174, 7811784, 4287447, -13924295, -30310973, -6749830, 67796411, 111607044, -17235948
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Links

Crossrefs

Cf. A077978.

Programs

  • GAP
    a:=[1,1,-1];; for n in [4..50] do a[n]:=a[n-1]-2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jul 02 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x+2*x^2+x^3) )); // G. C. Greubel, Jul 02 2019
  • Mathematica
    LinearRecurrence[{1,-2,-1}, {1,1,-1}, 50] (* or *) CoefficientList[ Series[1/(1-x+2*x^2+x^3), {x,0,50}], x] (* G. C. Greubel, Jul 02 2019 *)
  • PARI
    Vec(1/(1-x+2*x^2+x^3)+O(x^50)) \\ Charles R Greathouse IV, Sep 27 2012
  • Sage
    (1/(1-x+2*x^2+x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jul 02 2019

Formula

a(n) = (-1)^n * A077978(n). - G. C. Greubel, Jul 02 2019

A078049 Expansion of (1-x)/(1+x+2*x^2-x^3).

Original entry on oeis.org

1, -2, 0, 5, -7, -3, 22, -23, -24, 92, -67, -141, 367, -152, -723, 1394, -100, -3411, 5005, 1717, -15138, 16709, 15284, -63840, 49981, 92983, -256785, 120800, 485753, -984138, 133432, 2320597, -3571599, -936163, 10399958, -12099231, -9636848, 44235268, -37060803, -61046581, 179403455
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Links

Crossrefs

First differences of A077978.

Programs

Formula

G.f.: (-1 + x)/(-1 - x - 2*x^2 + x^3). - Michael De Vlieger, Jan 09 2020

Extensions

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A078021 Expansion of (1-x)/(1-x+2*x^2+x^3).

Original entry on oeis.org

1, 0, -2, -3, 1, 9, 10, -9, -38, -30, 55, 153, 73, -288, -587, -84, 1378, 2133, -539, -6183, -7238, 5667, 26326, 22230, -36089, -106875, -56927, 192912, 413641, 84744, -935450, -1518579, 267577, 4240185, 5223610, -3524337, -18211742, -16386678, 23561143, 74546241, 43810633, -128842992
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Links

Crossrefs

Cf. A077978.

Programs

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

Formula

a(n) = A077978(n) - A077978(n-1). - G. C. Greubel, Jun 29 2019
a(n) = a(n-1)-2*a(n-2)-a(n-3). - Wesley Ivan Hurt, Apr 26 2021

A276229 a(n+3) = -a(n+2) - 2*a(n+1) + a(n) with a(0)=0, a(1)=0, a(2)=1.

Original entry on oeis.org

0, 0, 1, -1, -1, 4, -3, -6, 16, -7, -31, 61, -6, -147, 220, 68, -655, 739, 639, -2772, 2233, 3950, -11188, 5521, 20805, -43035, 6946, 99929, -156856, -36056, 449697, -534441, -401009, 1919588, -1652011, -2588174, 7811784, -4287447, -13924295, 30310973
Offset: 0

Views

Author

G. C. Greubel, Aug 24 2016

Keywords

Comments

Essentially the same as A077978. - Georg Fischer, Oct 02 2018

Links

Crossrefs

Cf. A077939, A276228.

Programs

  • Magma
    I:=[0,0,1]; [n le 3 select I[n] else -Self(n-1)- 2*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Aug 25 2016
  • Mathematica
    LinearRecurrence[{-1, -2, 1}, {0, 0, 1}, 50]
CoefficientList[Series[x^2/(1 + x + 2 x^2 - x^3), {x, 0, 39}], x] (* Michael De Vlieger, Aug 25 2016 *)
  • PARI
    concat([0, 0], Vec(x^2/(1+x+2*x^2-x^3) + O(x^99))) \\ Altug Alkan, Aug 25 2016

Formula

G.f.: x^2/(1 + x + 2*x^2 - x^3).
Let P = (b-c)*(b-d), Q = (c-b)*(b-d), R = (d-b)*(d-c), (b, c, d) be the three roots of x^3 = 2*x^2 + x + 1, then a(n) = P^(-1)*b^(1-n) + Q^(-1)*c^(1-n) + R^(-1)*d^(1-n).
a(2*n) = -3*a(2*n-2) - 6*a(2*n-4) + a(2*n-6).
Showing 1-5 of 5 results.

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

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