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 1-3 of 3 results.

A376716 Expansion of (1 - x + x^2)/((1 - x + x^2)^2 - 4*x^2).

Original entry on oeis.org

1, 1, 4, 11, 27, 72, 189, 493, 1292, 3383, 8855, 23184, 60697, 158905, 416020, 1089155, 2851443, 7465176, 19544085, 51167077, 133957148, 350704367, 918155951, 2403763488, 6293134513, 16475640049, 43133785636, 112925716859, 295643364939, 774004377960
Offset: 0

Views

Author

Seiichi Manyama, Oct 02 2024

Keywords

Links

Crossrefs

Cf. A000302, A376717, A376718.
Cf. A182890.

Programs

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

Formula

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

A376718 Expansion of (1 - x + x^4)/((1 - x + x^4)^2 - 4*x^4).

Original entry on oeis.org

1, 1, 1, 1, 4, 11, 22, 37, 61, 114, 232, 467, 894, 1660, 3096, 5893, 11351, 21803, 41535, 78778, 149615, 285100, 544165, 1037963, 1977196, 3764056, 7167911, 13657244, 26027280, 49594720, 94481929, 179981485, 342872893, 653244245, 1244600984, 2371227307
Offset: 0

Views

Author

Seiichi Manyama, Oct 02 2024

Keywords

Links

Crossrefs

Cf. A000302, A376716, A376717.
Cf. A375283.

Programs

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

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 1-3 of 3 results.

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

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