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.

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).

A373908 Number of compositions of 7*n into parts 2 and 7.

Original entry on oeis.org

1, 1, 2, 9, 38, 136, 452, 1495, 5031, 17114, 58282, 198032, 671856, 2278870, 7731892, 26238839, 89047335, 302191369, 1025487338, 3479970844, 11809261583, 40074827170, 135994407483, 461498426696, 1566098800484, 5314568565096, 18035031128780, 61202027710656
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A373907, A373909, A373910, A373911, A373912.
Cf. A109961, A369840, A373904.
Cf. A369813.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(n+5*k, n-2*k));

Formula

a(n) = A369813(7*n).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+5*k,n-2*k).
a(n) = 7*a(n-1) - 20*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^2/(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).

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).
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.