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 1-6 of 6 results.

A373907 Number of compositions of 7*n into parts 1 and 7.

Original entry on oeis.org

1, 2, 10, 53, 264, 1294, 6349, 31200, 153366, 753836, 3705166, 18211117, 89508951, 439943336, 2162355196, 10628140702, 52238121106, 256754344524, 1261967164192, 6202664757387, 30486569842400, 149843813435961, 736493759087077, 3619922936674360
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A373908, A373909, A373910, A373911, A373912.
Cf. A005709, A369836.

Programs

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

Formula

a(n) = A005709(7*n).
a(n) = Sum_{k=0..n} binomial(n+6*k,n-k).
a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x/(1 - x)^6).

A373912 Number of compositions of 7*n into parts 6 and 7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1718, 3448, 6556, 12121, 22509, 43453, 89150, 193823, 436304, 989759, 2219064, 4869285, 10434412, 21900170, 45297211, 93054446, 191371581, 396480142, 830227401, 1756883373, 3746468095, 8017653633, 17151612398
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A373907, A373908, A373909, A373910, A373911.
Cf. A099099, A099131, A107025, A373913.
Cf. A017847.

Programs

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

Formula

a(n) = A017847(7*n).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-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.: 1/(1 - x - x^6/(1 - x)^6).

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).

A373910 Number of compositions of 7*n into parts 4 and 7.

Original entry on oeis.org

1, 1, 1, 1, 2, 9, 37, 121, 332, 808, 1837, 4113, 9497, 23091, 58462, 150129, 382810, 960520, 2373982, 5816480, 14230964, 34948927, 86295036, 213973997, 531470618, 1319411997, 3270186871, 8091796123, 20002405065, 49435009494, 122222402392, 302354237393
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A373907, A373908, A373909, A373911, A373912.
Cf. A369815.

Programs

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

Formula

a(n) = A369815(7*n).
a(n) = Sum_{k=0..floor(n/4)} binomial(n+3*k,n-4*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 34*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^4/(1 - x)^6).

A373911 Number of compositions of 7*n into parts 5 and 7.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 794, 1732, 3553, 7116, 14501, 31078, 70607, 166922, 399315, 946121, 2197582, 4998597, 11188280, 24835641, 55117511, 123036293, 276976136, 628285812, 1431723937, 3265884047, 7436635822, 16880558594, 38196652951, 86238054374
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A373907, A373908, A373909, A373910, A373912.
Cf. A369816.

Programs

  • Mathematica
    LinearRecurrence[{7,-21,35,-35,22,-7,1},{1,1,1,1,1,2,9},40] (* Harvey P. Dale, Oct 19 2024 *)
  • PARI
    a(n) = sum(k=0, n\5, binomial(n+2*k, n-5*k));

Formula

a(n) = A369816(7*n).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+2*k,n-5*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 22*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^5/(1 - x)^6).

A373904 a(n) = Sum_{k=0..floor(n/2)} binomial(n+4*k,n-2*k).

Original entry on oeis.org

1, 1, 2, 8, 30, 98, 303, 937, 2936, 9260, 29209, 91999, 289547, 911255, 2868341, 9029425, 28424456, 89478064, 281667368, 886657848, 2791106585, 8786123349, 27657838272, 87064092870, 274068969337, 862741412709, 2715822822365, 8549136056237, 26911817257385
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A011782, A034943, A109961, A369840, A373908.
Cf. A107025, A371125, A373905, A373906.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(n+4*k,n-2*k));

Formula

a(n) = 6*a(n-1) - 14*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: 1/(1 - x - x^2/(1 - x)^5).
Showing 1-6 of 6 results.

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