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.

A373933 Number of compositions of 7*n-1 into parts 6 and 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 17, 54, 175, 506, 1299, 3017, 6465, 13021, 25142, 47651, 91104, 180254, 374077, 810381, 1800140, 4019204, 8888489, 19322901, 41223071, 86520282, 179574728, 370946309, 767426451, 1597653852, 3354537225, 7101005320, 15118658953
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A369809, A373912, A373934, A373935, A373936, A373937.
Cf. A017847, A369849.

Programs

  • PARI
    a(n) = sum(k=0, n\6, binomial(n+k, n-1-6*k));

Formula

a(n) = A017847(7*n-1).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-1-6*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 6*a(n-6) + a(n-7).
G.f.: x*(1-x)^5/((1-x)^7 - x^6).
a(n) = A373934(n+1)-A373934(n). - R. J. Mathar, Jun 24 2024

A373934 Number of compositions of 7*n-2 into parts 6 and 7.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 29, 46, 100, 275, 781, 2080, 5097, 11562, 24583, 49725, 97376, 188480, 368734, 742811, 1553192, 3353332, 7372536, 16261025, 35583926, 76806997, 163327279, 342902007, 713848316, 1481274767, 3078928619, 6433465844, 13534471164
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A369809, A373912, A373933, A373935, A373936, A373937.
Cf. A017847, A369850.

Programs

  • PARI
    a(n) = sum(k=0, n\6, binomial(n+k, n-2-6*k));

Formula

a(n) = A017847(7*n-2).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-2-6*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 6*a(n-6) + a(n-7).
G.f.: x^2*(1-x)^4/((1-x)^7 - x^6).
a(n) = A373935(n+1)-A373935(n). - R. J. Mathar, Jun 24 2024

A373935 Number of compositions of 7*n-3 into parts 6 and 7.

Original entry on oeis.org

0, 0, 1, 4, 10, 20, 35, 56, 85, 131, 231, 506, 1287, 3367, 8464, 20026, 44609, 94334, 191710, 380190, 748924, 1491735, 3044927, 6398259, 13770795, 30031820, 65615746, 142422743, 305750022, 648652029, 1362500345, 2843775112, 5922703731, 12356169575
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A369809, A373912, A373933, A373934, A373936, A373937.
Cf. A017847.

Programs

  • PARI
    a(n) = sum(k=0, n\6, binomial(n+k, n-3-6*k));

Formula

a(n) = A017847(7*n-3).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-3-6*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 6*a(n-6) + a(n-7).
G.f.: x^3*(1-x)^3/((1-x)^7 - x^6).
a(n) = A373936(n+1)-A373936(n). - R. J. Mathar, Jun 24 2024

A373936 Number of compositions of 7*n-4 into parts 6 and 7.

Original entry on oeis.org

0, 0, 0, 1, 5, 15, 35, 70, 126, 211, 342, 573, 1079, 2366, 5733, 14197, 34223, 78832, 173166, 364876, 745066, 1493990, 2985725, 6030652, 12428911, 26199706, 56231526, 121847272, 264270015, 570020037, 1218672066, 2581172411, 5424947523, 11347651254
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A369809, A373912, A373933, A373934, A373935, A373937.
Cf. A017847, A369807.

Programs

  • PARI
    a(n) = sum(k=0, n\6, binomial(n+k, n-4-6*k));

Formula

a(n) = A017847(7*n-4).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-4-6*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 6*a(n-6) + a(n-7).
G.f.: x^4*(1-x)^2/((1-x)^7 - x^6).
a(n) = A373937(n+1)-A373937(n). - R. J. Mathar, Jun 24 2024
Showing 1-4 of 4 results.

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