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.

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).

A373302 Number of compositions of 6*n-2 into parts 1 and 6.

Original entry on oeis.org

1, 6, 27, 119, 533, 2402, 10829, 48804, 219925, 991044, 4465957, 20125051, 90690002, 408678475, 1841637299, 8299012941, 37398034921, 168527634148, 759439995404, 3422282105232, 15421909405056, 69496108849357, 313171930813206, 1411253951813003, 6359566489040219
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A099242, A371125, A373958, A373959, A373960.
Cf. A005708, A373928.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+3+5*k, n-1-k));

Formula

a(n) = A005708(6*n-2).
a(n) = Sum_{k=0..n} binomial(n+3+5*k,n-1-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.: x*(1-x)/((1-x)^6 - x).
a(n) = A099242(n-1) - A099242(n-2).

A373958 Number of compositions of 6*n-3 into parts 1 and 6.

Original entry on oeis.org

1, 5, 21, 92, 414, 1869, 8427, 37975, 171121, 771119, 3474913, 15659094, 70564951, 317988473, 1432958824, 6457375642, 29099021980, 131129599227, 590912361256, 2662842109828, 11999627299824, 54074199444301, 243675821963849, 1098082020999797, 4948312537227216
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A099242, A371125, A373302, A373959, A373960.
Cf. A005708.

Programs

  • Mathematica
    LinearRecurrence[{7,-15,20,-15,6,-1},{1,5,21,92,414,1869},30] (* Harvey P. Dale, Nov 04 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+2+5*k, n-1-k));

Formula

a(n) = A005708(6*n-3).
a(n) = Sum_{k=0..n} binomial(n+2+5*k,n-1-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.: x*(1-x)^2/((1-x)^6 - x).
a(n) = A373302(n) - A373302(n-1).

A373959 Number of compositions of 6*n-4 into parts 1 and 6.

Original entry on oeis.org

1, 4, 16, 71, 322, 1455, 6558, 29548, 133146, 599998, 2703794, 12184181, 54905857, 247423522, 1114970351, 5024416818, 22641646338, 102030577247, 459782762029, 2071929748572, 9336785189996, 42074572144477, 189601622519548, 854406199035948, 3850230516227419
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A099242, A371125, A373302, A373958, A373960.
Cf. A005708.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+1+5*k, n-1-k));

Formula

a(n) = A005708(6*n-4).
a(n) = Sum_{k=0..n} binomial(n+1+5*k,n-1-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.: x*(1-x)^3/((1-x)^6 - x).
a(n) = A373958(n) - A373958(n-1).

A373960 Number of compositions of 6*n-5 into parts 1 and 6.

Original entry on oeis.org

1, 3, 12, 55, 251, 1133, 5103, 22990, 103598, 466852, 2103796, 9480387, 42721676, 192517665, 867546829, 3909446467, 17617229520, 79388930909, 357752184782, 1612146986543, 7264855441424, 32737786954481, 147527050375071, 664804576516400, 2995824317191471
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A099242, A371125, A373302, A373958, A373959.
Cf. A005708.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+5*k, n-1-k));

Formula

a(n) = A005708(6*n-5).
a(n) = Sum_{k=0..n} binomial(n+5*k,n-1-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.: x*(1-x)^4/((1-x)^6 - x).
a(n) = A373959(n) - A373959(n-1).

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 1-6 of 6 results.

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

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