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 1-6 of 6 results.

A078019 Expansion of (1-x)/(1-x+2*x^2-x^3).

Original entry on oeis.org

1, 0, -2, -1, 3, 3, -4, -7, 4, 14, -1, -25, -9, 40, 33, -56, -82, 63, 171, -37, -316, -71, 524, 350, -769, -945, 943, 2064, -767, -3952, -354, 6783, 3539, -10381, -10676, 13625, 24596, -13330, -48897, 2359, 86823, 33208, -138079, -117672, 191694, 288959, -212101, -598325, 114836, 1099385, 271388
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Comments

With 1 prepended, and up to sign this is the q-deformation of 12/5. See Leclere and Morier-Genoud. - Michel Marcus, Jul 01 2021

Examples

			G.f. = 1 - 2*x^2 - x^3 + 3*x^4 + 3*x^5 - 4*x^6 - 7*x^7 + 4*x^8 + ...

Links

Crossrefs

Cf. A000931, A077954, A077979.

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
    CoefficientList[Series[(1-x)/(1-x+2x^2-x^3),{x,0,50}],x] (* or *) LinearRecurrence[{1,-2,1},{1,0,-2},51] (* Harvey P. Dale, Feb 18 2013 *)
  • PARI
    {a(n) = if( n<0, polcoeff( (1 - 2*x) / (1 - 2*x + x^2 - x^3) + x * O(x^-n), -n), polcoeff( (1 - x) / (1 - x + 2*x^2 - x^3) + x * O(x^n), n))} /* Michael Somos, Sep 18 2012 */
  • 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) = a(n-1) - 2*a(n-2) + a(n-3). - Michael Somos, Sep 18 2012
a(n) = -A000931(-2*n - 1). - Michael Somos, Sep 18 2012
G.f.: (1+x)/x^3 - 1/( Q(0) - x )/x^3 where Q(k) = 1 - x^2/(x^2*k - 1 )/Q(k+1) ; (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 23 2013
a(n) = (-1)^(n-1)*(A077979(n) + A077979(n-1)) = A077954(n) - A077954(n-1). - G. C. Greubel, Jun 29 2019

A360087 a(n) = Sum_{k=0..n} (-1)^k * binomial(3*k,n-k).

Original entry on oeis.org

1, -1, -2, 2, 6, -5, -17, 12, 48, -28, -135, 63, 378, -134, -1054, 259, 2927, -408, -8096, 280, 22305, 1551, -61210, -10638, 167310, 46683, -455489, -175852, 1234960, 612380, -3334215, -2031953, 8962498, 6523626, -23981046, -20445373, 63855135, 62900496
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Links

Crossrefs

Cf. A077979, A360088, A360089.
Cf. A099234.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(3*k, n-k));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/(1+x*(1+x)^3))

Formula

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

A360088 a(n) = Sum_{k=0..n} (-1)^k * binomial(4*k,n-k).

Original entry on oeis.org

1, -1, -3, 1, 13, 4, -49, -46, 165, 284, -476, -1417, 1003, 6220, -110, -24644, -14831, 88184, 113224, -278288, -619744, 715647, 2891977, -1036173, -12068353, -3381661, 45588556, 41600921, -154355594, -259984429, 448828716, 1305250324, -964837159, -5754843123
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Links

Crossrefs

Cf. A077979, A360087, A360089.
Cf. A099235.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(4*k, n-k));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/(1+x*(1+x)^4))

Formula

a(n) = -a(n-1) - 4*a(n-2) - 6*a(n-3) - 4*a(n-4) - a(n-5).
G.f.: 1/(1 + x*(1+x)^4).

A360089 a(n) = Sum_{k=0..n} (-1)^k * binomial(5*k,n-k).

Original entry on oeis.org

1, -1, -4, -1, 21, 29, -80, -244, 153, 1473, 836, -6920, -12220, 23209, 91213, -21511, -510680, -457965, 2210520, 4921941, -6271749, -33288595, -3876765, 173223185, 214943855, -682969376, -1912499375, 1498348275, 11882164650, 5332839025, -57402248250, -92821609874
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Links

Crossrefs

Cf. A077979, A360087, A360088.
Cf. A360090.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(5*k, n-k));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/(1+x*(1+x)^5))

Formula

a(n) = -a(n-1) - 5*a(n-2) - 10*a(n-3) - 10*a(n-4) - 5*a(n-5) - a(n-6).
G.f.: 1/(1 + x*(1+x)^5).

A077913 Expansion of 1/((1-x)*(1+x+2*x^2+x^3)).

Original entry on oeis.org

1, 0, -1, 1, 2, -2, -2, 5, 2, -9, 1, 16, -8, -24, 25, 32, -57, -31, 114, 6, -202, 77, 322, -273, -447, 672, 496, -1392, -271, 2560, -625, -4223, 2914, 6158, -7762, -7467, 16834, 5863, -32063, 3504, 54760, -29704, -83319, 87968, 108375, -200991, -103726, 397334, 11110, -702051, 282498, 1110495
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Links

Programs

  • GAP
    a:=[1,0,-1,1];; for n in [5..60] do a[n]:=-a[n-2]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jul 02 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1+x+2*x^2+x^3)) )); // G. C. Greubel, Jul 02 2019
  • Mathematica
    LinearRecurrence[{0,-1,1,1}, {1,0,-1,1}, 60] (* or *) CoefficientList[ Series[1/((1-x)*(1+x+2*x^2+x^3)), {x,0,60}], x] (* G. C. Greubel, Jul 02 2019 *)
  • PARI
    my(x='x+O('x^60)); Vec(1/((1-x)*(1+x+2*x^2+x^3))) \\ G. C. Greubel, Jul 02 2019
  • Sage
    (1/((1-x)*(1+x+2*x^2+x^3))).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Jul 02 2019

Formula

G.f.: 1-x^2/(U(0)+x^2) where U(k)= 1 + (1+x)*x/( 1 - (1+x)*x/((1+x)*x + 1/U(k+1))) ; (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 24 2012
5*a(n) = 1 + 4*A077979(n) + 3*A077979(n-1) + A077979(n-2). - R. J. Mathar, Jul 10 2013

A078051 Expansion of (1-x)/(1+x+2*x^2+x^3).

Original entry on oeis.org

1, -2, 0, 3, -1, -5, 4, 7, -10, -8, 21, 5, -39, 8, 65, -42, -96, 115, 119, -253, -100, 487, -34, -840, 421, 1293, -1295, -1712, 3009, 1710, -6016, -413, 10735, -3893, -17164, 14215, 24006, -35272, -26955, 73493, 15689, -135720, 30849, 224902, -150880, -329773, 406631, 403795, -887284, -326937
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Links

Crossrefs

First differences of A077979.

Programs

  • Mathematica
    LinearRecurrence[{-1,-2,-1},{1,-2,0},50] (* Harvey P. Dale, Aug 18 2014 *)
  • PARI
    Vec((1-x)/(1+x+2*x^2+x^3) + O(x^50)) \\ Jinyuan Wang, Apr 07 2020
Showing 1-6 of 6 results.

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

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