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 1-4 of 4 results.

A375317 Expansion of (1 + x)^2/(1 - x^2*(1 + x)^3).

Original entry on oeis.org

1, 2, 2, 5, 11, 18, 34, 68, 126, 235, 450, 851, 1601, 3032, 5739, 10838, 20489, 38752, 73252, 138472, 261813, 494973, 935737, 1769080, 3344567, 6323023, 11953991, 22599701, 42725841, 80775310, 152709940, 288705927, 545813094, 1031887518, 1950836005, 3688154521
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2024

Keywords

Links

Crossrefs

Cf. A116090, A375315.

Programs

  • Mathematica
    CoefficientList[Series[(1+x)^2/(1-x^2(1+x)^3),{x,0,40}],x] (* or *) LinearRecurrence[{0,1,3,3,1},{1,2,2,5,11},40] (* Harvey P. Dale, Mar 25 2025 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec((1+x)^2/(1-x^2*(1+x)^3))
  • PARI
    a(n) = sum(k=0, n\2, binomial(3*k+2, n-2*k));

Formula

a(n) = a(n-2) + 3*a(n-3) + 3*a(n-4) + a(n-5).
a(n) = Sum_{k=0..floor(n/2)} binomial(3*k+2,n-2*k).
a(n) = A375315(n) + A375315(n-1).

A375316 Expansion of (1 + x)/(1 - x^2*(1 + x)^4).

Original entry on oeis.org

1, 1, 1, 5, 11, 19, 42, 98, 205, 429, 936, 2024, 4316, 9260, 19949, 42841, 91917, 197485, 424331, 911255, 1957086, 4203998, 9029949, 19394681, 41657808, 89478064, 192189304, 412801176, 886657081, 1904452689, 4090567673, 8786123349, 18871714923, 40534539675
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2024

Keywords

Links

Crossrefs

Cf. A093040, A375315.
Cf. A375314.

Programs

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

Formula

a(n) = a(n-2) + 4*a(n-3) + 6*a(n-4) + 4*a(n-5) + a(n-6).
a(n) = Sum_{k=0..floor(n/2)} binomial(4*k+1,n-2*k).
a(n) = A375314(n) + A375314(n-1).

A375364 Expansion of 1/( (1 + x) * (1 - x^2*(1 + x)^3) ).

Original entry on oeis.org

1, -1, 2, 1, 3, 4, 12, 17, 35, 67, 127, 234, 451, 850, 1602, 3031, 5740, 10837, 20490, 38751, 73253, 138471, 261814, 494972, 935738, 1769079, 3344568, 6323022, 11953992, 22599700, 42725842, 80775309, 152709941, 288705926, 545813095, 1031887517, 1950836006
Offset: 0

Views

Author

Seiichi Manyama, Aug 13 2024

Keywords

Links

Crossrefs

Cf. A375315, A375317, A375365.

Programs

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

Formula

a(n) = -a(n-1) + a(n-2) + 4*a(n-3) + 6*a(n-4) + 4*a(n-5) + a(n-6).
a(n) = Sum_{k=0..floor(n/2)} binomial(3*k-1,n-2*k).
a(n) = A375365(n) + A375365(n-1).

A375365 Expansion of 1/( (1 + x)^2 * (1 - x^2*(1 + x)^3) ).

Original entry on oeis.org

1, -2, 4, -3, 6, -2, 14, 3, 32, 35, 92, 142, 309, 541, 1061, 1970, 3770, 7067, 13423, 25328, 47925, 90546, 171268, 323704, 612034, 1157045, 2187523, 4135499, 7818493, 14781207, 27944635, 52830674, 99879267, 188826659, 356986436, 674901081, 1275934925, 2412219595
Offset: 0

Views

Author

Seiichi Manyama, Aug 13 2024

Keywords

Links

Crossrefs

Cf. A375315, A375317, A375364.

Programs

  • Mathematica
    CoefficientList[Series[1/((1+x)^2(1-x^2(1+x)^3)),{x,0,40}],x] (* or *) LinearRecurrence[{-2,0,5,10,10,5,1},{1,-2,4,-3,6,-2,14},40] (* Harvey P. Dale, Aug 07 2025 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/((1+x)^2*(1-x^2*(1+x)^3)))
  • PARI
    a(n) = sum(k=0, n\2, binomial(3*k-2, n-2*k));

Formula

a(n) = -2*a(n-1) + 5*a(n-3) + 10*a(n-4) + 10*a(n-5) + 5*a(n-6) + a(n-7).
a(n) = Sum_{k=0..floor(n/2)} binomial(3*k-2,n-2*k).
Showing 1-4 of 4 results.

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