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 1-5 of 5 results.

A369806 Expansion of 1/(1 - x^3/(1-x)^7).

Original entry on oeis.org

1, 0, 0, 1, 7, 28, 85, 224, 567, 1485, 4117, 11802, 33909, 96182, 269402, 750275, 2090728, 5845015, 16384908, 45973701, 128944042, 361364501, 1012168575, 2834690172, 7939970075, 22244001961, 62323608147, 174620915138, 489240430938, 1370662332271, 3839992876850
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Comments

Number of compositions of 7*n-3 into parts 3 and 7.

Links

Crossrefs

Cf. A099253, A369805, A369807, A369808, A369809.
Cf. A369814.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/(1-x^3/(1-x)^7))
  • PARI
    a(n) = sum(k=0, n\3, binomial(n-1+4*k, n-3*k));

Formula

a(n) = A369814(7*n-3) for n > 0.
a(n) = 7*a(n-1) - 21*a(n-2) + 36*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 7.
a(n) = Sum_{k=0..floor(n/3)} binomial(n-1+4*k,n-3*k).

A369813 Expansion of 1/(1 - x^2 - x^7).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 4, 6, 7, 7, 11, 9, 16, 13, 22, 20, 29, 31, 38, 47, 51, 69, 71, 98, 102, 136, 149, 187, 218, 258, 316, 360, 452, 509, 639, 727, 897, 1043, 1257, 1495, 1766, 2134, 2493, 3031, 3536, 4288, 5031, 6054, 7165, 8547, 10196, 12083, 14484, 17114, 20538, 24279, 29085, 34475
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2024

Keywords

Comments

Number of compositions of n into parts 2 and 7.

Links

Crossrefs

Cf. A005709, A017847, A369814, A369815, A369816.
Cf. A007380.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-x^2-x^7),{x,0,80}],x] (* or *) LinearRecurrence[{0,1,0,0,0,0,1},{1,0,1,0,1,0,1},80] (* Harvey P. Dale, Dec 04 2024 *)
  • PARI
    my(N=70, x='x+O('x^N)); Vec(1/(1-x^2-x^7))
  • PARI
    a(n) = sum(k=0, n\7, ((n-5*k)%2==0)*binomial((n-5*k)/2, k));

Formula

a(n) = a(n-2) + a(n-7).
a(n) = A007380(n-7) for n >= 8.

A369815 Expansion of 1/(1 - x^4 - x^7).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 3, 1, 0, 3, 4, 1, 1, 6, 5, 1, 4, 10, 6, 2, 10, 15, 7, 6, 20, 21, 9, 16, 35, 28, 15, 36, 56, 37, 31, 71, 84, 52, 67, 127, 121, 83, 138, 211, 173, 150, 265, 332, 256, 288, 476, 505, 406, 553, 808, 761, 694, 1029, 1313, 1167, 1247, 1837, 2074, 1861, 2276, 3150, 3241
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2024

Keywords

Comments

Number of compositions of n into parts 4 and 7.

Links

Crossrefs

Cf. A005709, A017847, A369813, A369814, A369816.

Programs

  • PARI
    my(N=80, x='x+O('x^N)); Vec(1/(1-x^4-x^7))
  • PARI
    a(n) = sum(k=0, n\7, ((n-3*k)%4==0)*binomial((n-3*k)/4, k));

Formula

a(n) = a(n-4) + a(n-7).

A369816 Expansion of 1/(1 - x^5 - x^7).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 3, 1, 1, 4, 0, 6, 1, 4, 5, 1, 10, 1, 10, 6, 5, 15, 2, 20, 7, 15, 21, 7, 35, 9, 35, 28, 22, 56, 16, 70, 37, 57, 84, 38, 126, 53, 127, 121, 95, 210, 91, 253, 174, 222, 331, 186, 463, 265, 475, 505, 408, 794, 451, 938, 770, 883, 1299, 859, 1732, 1221
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2024

Keywords

Comments

Number of compositions of n into parts 5 and 7.

Links

Crossrefs

Cf. A005709, A017847, A369813, A369814, A369815.

Programs

  • PARI
    my(N=80, x='x+O('x^N)); Vec(1/(1-x^5-x^7))
  • PARI
    a(n) = sum(k=0, n\7, ((n-2*k)%5==0)*binomial((n-2*k)/5, k));

Formula

a(n) = a(n-5) + a(n-7).
G.f.: 1/((1-x+x^2)*(1+x-x^3-x^4-x^5)). - R. J. Mathar, Jul 03 2024

A373909 Number of compositions of 7*n into parts 3 and 7.

Original entry on oeis.org

1, 1, 1, 2, 9, 37, 122, 346, 913, 2398, 6515, 18317, 52226, 148408, 417810, 1168085, 3258813, 9103828, 25488736, 71462437, 200406479, 561770980, 1573939555, 4408629727, 12348599802, 34592601763, 96916209910, 271537125048, 760777555986, 2131439888257
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A373907, A373908, A373910, A373911, A373912.
Cf. A369814.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n+4*k, n-3*k));

Formula

a(n) = A369814(7*n).
a(n) = Sum_{k=0..floor(n/3)} binomial(n+4*k,n-3*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 36*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^3/(1 - x)^6).
Showing 1-5 of 5 results.

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