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 1-7 of 7 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).

A369840 Number of compositions of 5*n into parts 2 and 5.

Original entry on oeis.org

1, 1, 2, 7, 23, 68, 194, 555, 1601, 4633, 13404, 38752, 112004, 323728, 935737, 2704817, 7818464, 22599701, 65325542, 188826693, 545813094, 1577700612, 4560424135, 13182138184, 38103641048, 110140512968, 318366757185, 920255312908, 2660044812499, 7688994894381
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A369803, A369842, A369843, A369844.
Cf. A099131, A369836, A369845.
Cf. A001687.

Programs

  • Mathematica
    LinearRecurrence[{5, -9, 10, -5, 1}, {1, 1, 2, 7, 23}, 50] (* Paolo Xausa, Mar 15 2024 *)
  • PARI
    a(n) = sum(k=0, n\2, binomial(n+3*k, n-2*k));

Formula

a(n) = A001687(5*n+1).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+3*k,n-2*k).
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (1-x)^4/((1-x)^5 - x^2).
a(n) = A369843(n) - A369843(n-1). - R. J. Mathar, Feb 14 2024

A371125 Number of compositions of 6*n into parts 1 and 6.

Original entry on oeis.org

1, 2, 9, 43, 196, 882, 3970, 17887, 80608, 363254, 1636944, 7376591, 33241289, 149795989, 675029164, 3041899638, 13707783053, 61771701389, 278363253873, 1254394801761, 5652708454881, 25472931513057, 114789263420590, 517277526141329, 2331019740675071
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A052544, A055988, A369836, A373907, A373890.
Cf. A107025, A373904, A373905, A373906.
Cf. A005708.

Programs

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

Formula

a(n) = A005708(6*n).
a(n) = Sum_{k=0..n} binomial(n+5*k,n-k).
a(n) = 7*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: 1/(1 - x - x/(1 - x)^5).

A369837 Number of compositions of 5*n-2 into parts 1 and 5.

Original entry on oeis.org

1, 5, 20, 80, 325, 1326, 5411, 22076, 90061, 367411, 1498887, 6114853, 24946129, 101770120, 415180936, 1693770328, 6909898016, 28189589705, 115002126790, 469162173146, 1913991948274, 7808313175575, 31854760257925, 129954540535600, 530161974821876
Offset: 1

Views

Author

Seiichi Manyama, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A079675, A369836, A369838, A369839.
Cf. A003520, A055990.

Programs

  • Mathematica
    LinearRecurrence[{6, -10, 10, -5, 1}, {1, 5, 20, 80, 325}, 50] (* Paolo Xausa, Mar 15 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+2+4*k, n-1-k));

Formula

a(n) = A003520(5*n-2).
a(n) = Sum_{k=0..n} binomial(n+2+4*k,n-1-k).
a(n) = 6*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)/((1-x)^5 - x).

A369838 Number of compositions of 5*n-3 into parts 1 and 5.

Original entry on oeis.org

1, 4, 15, 60, 245, 1001, 4085, 16665, 67985, 277350, 1131476, 4615966, 18831276, 76823991, 313410816, 1278589392, 5216127688, 21279691689, 86812537085, 354160046356, 1444829775128, 5894321227301, 24046447082350, 98099780277675, 400207434286276, 1632684497403029
Offset: 1

Views

Author

Seiichi Manyama, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A079675, A369836, A369837, A369839.
Cf. A003520.

Programs

  • Mathematica
    LinearRecurrence[{6, -10, 10, -5, 1}, {1, 4, 15, 60, 245}, 50] (* Paolo Xausa, Mar 15 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+1+4*k, n-1-k));

Formula

a(n) = A003520(5*n-3).
a(n) = Sum_{k=0..n} binomial(n+1+4*k,n-1-k).
a(n) = 6*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)^2/((1-x)^5 - x).

A369839 Number of compositions of 5*n-4 into parts 1 and 5.

Original entry on oeis.org

1, 3, 11, 45, 185, 756, 3084, 12580, 51320, 209365, 854126, 3484490, 14215310, 57992715, 236586825, 965178576, 3937538296, 16063564001, 65532845396, 267347509271, 1090669728772, 4449491452173, 18152125855049, 74053333195325, 302107654008601, 1232477063116753
Offset: 1

Views

Author

Seiichi Manyama, Feb 03 2024

Keywords

Links

Crossrefs

Cf. A079675, A369836, A369837, A369838.
Cf. A003520.

Programs

  • Mathematica
    LinearRecurrence[{6, -10, 10, -5, 1}, {1, 3, 11, 45, 185}, 50] (* Paolo Xausa, Mar 15 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+4*k, n-1-k));

Formula

a(n) = A003520(5*n-4).
a(n) = Sum_{k=0..n} binomial(n+4*k,n-1-k).
a(n) = 6*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)^3/((1-x)^5 - x).

A373890 Number of compositions of 8*n into parts 1 and 8.

Original entry on oeis.org

1, 2, 11, 64, 345, 1824, 9661, 51284, 272333, 1445995, 7677250, 40760798, 216412235, 1149004281, 6100444144, 32389272248, 171965334801, 913020717480, 4847528344990, 25737127996244, 136646907481155, 725503534206186, 3851937726561990, 20451208781128462
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A052544, A055988, A369836, A371125, A373907.
Cf. A005710.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+7*k, n-k));

Formula

a(n) = A005710(8*n).
a(n) = Sum_{k=0..n} binomial(n+7*k,n-k).
a(n) = 9*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
G.f.: 1/(1 - x - x/(1 - x)^7).
Showing 1-7 of 7 results.

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