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 1-9 of 9 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).

A376723 Expansion of 1/((1 - x^2 - x^3)^2 - 4*x^5).

Original entry on oeis.org

1, 0, 2, 2, 3, 10, 7, 28, 33, 64, 132, 170, 408, 578, 1119, 2002, 3194, 6310, 10021, 18666, 32353, 55450, 101443, 170672, 308744, 534820, 935936, 1663892, 2872669, 5111652, 8898082, 15641802, 27538647, 48049562, 84813451, 148219128, 260572901, 457451088
Offset: 0

Views

Author

Seiichi Manyama, Oct 02 2024

Keywords

Links

Crossrefs

Cf. A182890, A376724, A376725.
Cf. A376726, A376729.
Cf. A375278.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/((1-x^2-x^3)^2-4*x^5))
  • PARI
    a(n) = sum(k=0, n\2, binomial(2*k+2, 2*n-4*k+1))/2;

Formula

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6).
a(n) = (1/2) * Sum_{k=0..floor(n/2)} binomial(2*k+2,2*n-4*k+1).

A375283 Expansion of 1/((1 - x - x^4)^2 - 4*x^5).

Original entry on oeis.org

1, 2, 3, 4, 7, 16, 35, 68, 122, 220, 417, 816, 1588, 3028, 5707, 10784, 20547, 39322, 75150, 143144, 272212, 517990, 987005, 1881824, 3586808, 6832874, 13013780, 24789200, 47229672, 89991518, 171459667, 326651952, 622295173, 1185547900, 2258689217, 4303264572
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2024

Keywords

Links

Crossrefs

Cf. A182890, A375278, A375285.
Cf. A246883, A375282.

Programs

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

Formula

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

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).

A375285 Expansion of 1/((1 - x - x^5)^2 - 4*x^6).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 17, 36, 69, 120, 196, 320, 547, 980, 1786, 3216, 5661, 9804, 16932, 29472, 51820, 91602, 161767, 284424, 498103, 871150, 1525380, 2676544, 4703158, 8265354, 14514236, 25464576, 44656997, 78324398, 137430720, 241225072, 423451668, 743244866
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2024

Keywords

Links

Crossrefs

Cf. A182890, A375278, A375283.
Cf. A246884.

Programs

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

Formula

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

A375288 Expansion of 1/((1 - x + x^3)^2 + 4*x^4).

Original entry on oeis.org

1, 2, 3, 2, -5, -22, -50, -74, -47, 122, 544, 1230, 1816, 1144, -3029, -13416, -30267, -44578, -27815, 75170, 330874, 744780, 1094243, 676196, -1865344, -8160100, -18326608, -26859600, -16435947, 46284926, 201243559, 450953386, 659291863, 399432970, -1148383866
Offset: 0

Views

Author

Seiichi Manyama, Aug 10 2024

Keywords

Links

Crossrefs

Cf. A375255, A375290.
Cf. A375278, A375292.

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, (-1)^k*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)} (-1)^k * binomial(2*n-4*k+2,2*k+1).

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).

A387602 a(n) = (1/2) * Sum_{k=0..floor(n/3)} 2^(n-2*k) * binomial(2*n-4*k+2,2*k+1).

Original entry on oeis.org

1, 4, 12, 36, 120, 416, 1420, 4768, 15968, 53664, 180736, 608640, 2048336, 6891968, 23191104, 78044352, 262644608, 883866624, 2974400960, 10009502720, 33684265984, 113355412480, 381467226112, 1283724873728, 4320028764416, 14537889756160, 48923344206848
Offset: 0

Views

Author

Seiichi Manyama, Sep 02 2025

Keywords

Links

Crossrefs

Cf. A375278, A387604.
Cf. A387510.

Programs

  • Magma
    [&+[2^(n-2*k)* Binomial(2*n-4*k+2, 2*k+1)/2: k in [0..Floor (n/3)]]: n in [0..35]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[2^(n-2*k)*Binomial[2*n-4*k+2, 2*k+1]/2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 03 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, 2^(n-2*k)*binomial(2*n-4*k+2, 2*k+1))/2;

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A387510.
G.f.: 1/((1-2*x-2*x^3)^2 - 16*x^4).
a(n) = 4*a(n-1) - 4*a(n-2) + 4*a(n-3) + 8*a(n-4) - 4*a(n-6).

A387604 a(n) = (1/2) * Sum_{k=0..floor(n/3)} 3^(n-2*k) * binomial(2*n-4*k+2,2*k+1).

Original entry on oeis.org

1, 6, 27, 114, 495, 2214, 9990, 44982, 201933, 905526, 4061016, 18217710, 81735156, 366712272, 1645244379, 7381235808, 33115172733, 148568241906, 666539094105, 2990373257970, 13416063062094, 60190050847500, 270037644213267, 1211501390490972, 5435300133382176
Offset: 0

Views

Author

Seiichi Manyama, Sep 02 2025

Keywords

Links

Crossrefs

Cf. A375278, A387602.
Cf. A387513.

Programs

  • Magma
    [&+[3^(n-2*k)* Binomial(2*n-4*k+2, 2*k+1)/2: k in [0..Floor (n/3)]]: n in [0..35]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[3^(n-2*k)*Binomial[2*n-4*k+2, 2*k+1]/2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 03 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, 3^(n-2*k)*binomial(2*n-4*k+2, 2*k+1))/2;

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A387513.
G.f.: 1/((1-3*x-3*x^3)^2 - 36*x^4).
a(n) = 6*a(n-1) - 9*a(n-2) + 6*a(n-3) + 18*a(n-4) - 9*a(n-6).
Showing 1-9 of 9 results.

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