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 1-7 of 7 results.

A375278 Expansion of 1/((1 - x - x^3)^2 - 4*x^4).

Original entry on oeis.org

1, 2, 3, 6, 15, 34, 70, 146, 317, 690, 1480, 3162, 6788, 14608, 31395, 67392, 144701, 310854, 667793, 1434310, 3080542, 6616676, 14212315, 30526804, 65567936, 140832740, 302495240, 649730544, 1395554885, 2997508382, 6438345511, 13828920758, 29703127299
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2024

Keywords

Links

Crossrefs

Cf. A182890, A375283, A375285.
Cf. A246840, A375279.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/((1-x-x^3)^2-4*x^4))
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-4*k+2, 2*k+1))/2;

Formula

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) - a(n-6).
a(n) = (1/2) * Sum_{k=0..floor(n/3)} binomial(2*n-4*k+2,2*k+1).

A375282 Expansion of (1 - x - x^4)/((1 - x - x^4)^2 - 4*x^5).

Original entry on oeis.org

1, 1, 1, 1, 2, 7, 16, 29, 47, 82, 162, 331, 650, 1220, 2262, 4261, 8175, 15747, 30121, 57210, 108521, 206456, 393865, 751675, 1432772, 2728076, 5193901, 9893596, 18853664, 35928972, 68454369, 130403085, 248413549, 473261209, 901681650, 1717923403, 3272944760
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2024

Keywords

Links

Crossrefs

Cf. A375279.

Programs

  • Mathematica
    CoefficientList[Series[(1-x-x^4)/((1-x-x^4)^2-4x^5),{x,0,40}],x] (* or *) LinearRecurrence[{2,-1,0,2,2,0,0,-1},{1,1,1,1,2,7,16,29},40] (* Harvey P. Dale, May 24 2025 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec((1-x-x^4)/((1-x-x^4)^2-4*x^5))
  • PARI
    a(n) = sum(k=0, n\4, binomial(2*n-6*k, 2*k));

Formula

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-4) + 2*a(n-5) - a(n-8).
a(n) = Sum_{k=0..floor(n/4)} binomial(2*n-6*k,2*k).

A376717 Expansion of (1 - x + x^3)/((1 - x + x^3)^2 - 4*x^3).

Original entry on oeis.org

1, 1, 1, 4, 11, 22, 42, 91, 205, 443, 936, 1999, 4316, 9300, 19949, 42785, 91917, 197548, 424331, 911218, 1957086, 4203927, 9029949, 19395031, 41657808, 89477119, 192189304, 412803240, 886657081, 1904448737, 4090567673, 8786130132, 18871714923, 40534529294
Offset: 0

Views

Author

Seiichi Manyama, Oct 02 2024

Keywords

Links

Crossrefs

Cf. A000302, A376716, A376718.
Cf. A375278, A375279.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec((1-x+x^3)/((1-x+x^3)^2-4*x^3))
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-4*k+1, 2*k));

Formula

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) - a(n-6).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-4*k+1,2*k).

A375284 Expansion of (1 - x - x^5)/((1 - x - x^5)^2 - 4*x^6).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 16, 29, 46, 68, 107, 191, 364, 686, 1234, 2125, 3596, 6148, 10754, 19132, 34121, 60361, 105725, 184207, 321227, 562628, 989397, 1742190, 3064093, 5377732, 9424960, 16515877, 28964243, 50840968, 89280116, 156762020, 275136201, 482728432
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2024

Keywords

Links

Crossrefs

Cf. A108479, A375279, A375282.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec((1-x-x^5)/((1-x-x^5)^2-4*x^6))
  • PARI
    a(n) = sum(k=0, n\5, binomial(2*n-8*k, 2*k));

Formula

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-5) + 2*a(n-6) - a(n-10).
a(n) = Sum_{k=0..floor(n/5)} binomial(2*n-8*k,2*k).

A375289 Expansion of (1 - x + x^3)/((1 - x + x^3)^2 + 4*x^4).

Original entry on oeis.org

1, 1, 1, 0, -5, -14, -26, -29, 5, 119, 348, 639, 708, -128, -2943, -8571, -15707, -17340, 3347, 72718, 211126, 386091, 424633, -87173, -1796760, -5200513, -9490312, -10398336, 2263553, 44394265, 128099033, 233273880, 254623403, -58615334, -1096863450, -3155300397
Offset: 0

Views

Author

Seiichi Manyama, Aug 10 2024

Keywords

Links

Crossrefs

Cf. A375279.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec((1-x+x^3)/((1-x+x^3)^2+4*x^4))
  • PARI
    a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-4*k, 2*k));

Formula

a(n) = 2*a(n-1) - a(n-2) - 2*a(n-3) - 2*a(n-4) - a(n-6).
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-4*k,2*k).

A375308 a(n) = Sum_{k=0..floor(2*n/3)} binomial(4*n-4*k,2*k).

Original entry on oeis.org

1, 1, 7, 30, 137, 644, 2936, 13625, 62701, 289547, 1335378, 6161329, 28424456, 131135696, 604991601, 2791106585, 12876698159, 59406240678, 274068969337, 1264408966284, 5833313285128, 26911817257385, 124156868897413, 572794023175795, 2642568194952474
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2024

Keywords

Links

Crossrefs

Cf. A108479, A375279, A375307, A375314.

Programs

  • PARI
    a(n) = sum(k=0, 2*n\3, binomial(4*n-4*k, 2*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec((1-x-6*x^2-x^3)/((1-x+2*x^2-x^3)^2-16*x^2))

Formula

a(n) = A375279(2*n).
a(n) = A375314(2*n).
a(n) = 2*a(n-1) + 11*a(n-2) + 6*a(n-3) - 6*a(n-4) + 4*a(n-5) - a(n-6).
G.f.: (1 - x - 6*x^2 - x^3)/((1 - x + 2*x^2 - x^3)^2 - 16*x^2).

A376785 Expansion of (1 + x - x^3)/((1 + x - x^3)^2 - 4*x).

Original entry on oeis.org

1, 3, 5, 8, 19, 46, 98, 201, 429, 937, 2024, 4325, 9260, 19916, 42841, 91999, 197485, 424160, 911255, 1957402, 4203998, 9029425, 19394681, 41658577, 89478064, 192188361, 412801176, 886657848, 1904452689, 4090568027, 8786123349, 18871711384, 40534539675, 87064092870
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2024

Keywords

Links

Crossrefs

Cf. A099511, A376786.
Cf. A375278, A375279, A376717.

Programs

  • Mathematica
    CoefficientList[Series[(1+x-x^3)/((1+x-x^3)^2-4x),{x,0,40}],x] (* or *) LinearRecurrence[{2,-1,2,2,0,-1},{1,3,5,8,19,46},40] (* Harvey P. Dale, Jun 29 2025 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec((1+x-x^3)/((1+x-x^3)^2-4*x))
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-4*k+1, 2*k+1));

Formula

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) - a(n-6).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-4*k+1,2*k+1).
Showing 1-7 of 7 results.

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