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.

Previous Showing 11-17 of 17 results.

A375279 Expansion of (1 - x - x^3)/((1 - x - x^3)^2 - 4*x^4).

Original entry on oeis.org

1, 1, 1, 2, 7, 16, 30, 61, 137, 303, 644, 1365, 2936, 6340, 13625, 29209, 62701, 134758, 289547, 621816, 1335378, 2868341, 6161329, 13233947, 28424456, 61052489, 131135696, 281667368, 604991601, 1299458257, 2791106585, 5995020362, 12876698159, 27657838272
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2024

Keywords

Links

Crossrefs

Cf. A108479, A375278.

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, 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)} binomial(2*n-4*k,2*k).

A376730 Expansion of (1 - x^3 - x^4)/((1 - x^3 - x^4)^2 - 4*x^7).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 6, 1, 1, 15, 15, 2, 28, 70, 29, 46, 210, 211, 111, 496, 925, 586, 1067, 3005, 3123, 2821, 8100, 13024, 11068, 20385, 44068, 48604, 57325, 129261, 192224, 200585, 358806, 662117, 781433, 1055567, 2050819, 2941702, 3524140, 6067682, 10169037
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2024

Keywords

Links

Crossrefs

Cf. A108479, A376729, A376731.
Cf. A376724, A376727.

Programs

  • PARI
    my(N=50, x='x+O('x^N)); Vec((1-x^3-x^4)/((1-x^3-x^4)^2-4*x^7))
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*k, 2*n-6*k));

Formula

a(n) = 2*a(n-3) + 2*a(n-4) - a(n-6) + 2*a(n-7) - a(n-8).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,2*n-6*k).

A376731 Expansion of (1 - x^4 - x^5)/((1 - x^4 - x^5)^2 - 4*x^9).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 1, 6, 1, 0, 1, 15, 15, 1, 1, 28, 70, 28, 2, 45, 210, 210, 46, 67, 495, 924, 496, 157, 1002, 3003, 3004, 1121, 1911, 8009, 12871, 8161, 4880, 18684, 43760, 43948, 23409, 41820, 126124, 184988, 133285, 113373, 324616, 647112, 657273, 454366
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2024

Keywords

Links

Crossrefs

Cf. A108479, A376729, A376730.
Cf. A376725, A376728.

Programs

  • PARI
    my(N=60, x='x+O('x^N)); Vec((1-x^4-x^5)/((1-x^4-x^5)^2-4*x^9))
  • PARI
    a(n) = sum(k=0, n\4, binomial(2*k, 2*n-8*k));

Formula

a(n) = 2*a(n-4) + 2*a(n-5) - a(n-8) + 2*a(n-9) - a(n-10).
a(n) = Sum_{k=0..floor(n/4)} binomial(2*k,2*n-8*k).

A375275 Expansion of (1 - x + x^2)/(1 - 2*x + 3*x^2 + 2*x^3 + x^4).

Original entry on oeis.org

1, 1, 0, -5, -13, -12, 25, 117, 196, 3, -841, -2200, -2079, 4121, 19720, 33435, 1547, -140772, -372775, -359763, 678796, 3323203, 5702319, 437200, -23557759, -63154959, -62213360, 111716475, 559940707, 972313668, 103585625, -3941367643, -10698060204
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2024

Keywords

Links

Crossrefs

Cf. A108479, A108480, A108484.
Cf. A375255.

Programs

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

Formula

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

A375307 a(n) = Sum_{k=0..floor(3*n/5)} binomial(3*n-3*k,2*k).

Original entry on oeis.org

1, 1, 4, 16, 52, 194, 685, 2452, 8771, 31327, 112004, 400285, 1430710, 5113647, 18277014, 65325542, 233485250, 834519021, 2982723523, 10660798289, 38103641048, 136189372297, 486765693153, 1739789499591, 6218325456983, 22225431015537, 79437750107600
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2024

Keywords

Links

Crossrefs

Cf. A108479, A116090, A375308.

Programs

  • PARI
    a(n) = sum(k=0, 3*n\5, binomial(3*n-3*k, 2*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec((1-x-3*x^2)/(1-2*x-5*x^2-3*x^3+3*x^4-x^5))

Formula

a(n) = A116090(2*n).
a(n) = 2*a(n-1) + 5*a(n-2) + 3*a(n-3) - 3*a(n-4) + a(n-5).
G.f.: (1 - x - 3*x^2)/(1 - 2*x - 5*x^2 - 3*x^3 + 3*x^4 - x^5).

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).
Previous Showing 11-17 of 17 results.

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