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 1-10 of 11 results. Next

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).

A369794 Expansion of 1/(1 - x^5/(1-x)^6).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 6, 21, 56, 126, 253, 474, 870, 1651, 3367, 7372, 16762, 38183, 85290, 185573, 394555, 826752, 1724816, 3613968, 7642004, 16313856, 35052905, 75487110, 162349105, 348018300, 743376838, 1583718457, 3370144462, 7173308802, 15285181447
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Comments

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

Links

Crossrefs

Cf. A095263, A290998, A368475.
Cf. A000389, A099242, A017837, A107025.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/(1-x^5/(1-x)^6))

Formula

a(n) = A107025(n)-A107025(n-1). First differences of A107025.
a(n) = A017837(6*n-5) = Sum_{k=0..floor((6*n-5)/5)} binomial(k,6*n-5-5*k) for n > 0.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 7*a(n-5) - a(n-6) for n > 6.
a(n) = Sum_{k=0..floor(n/5)} binomial(n-1+k,n-5*k).

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).

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).

A373905 a(n) = Sum_{k=0..floor(n/3)} binomial(n+3*k,n-3*k).

Original entry on oeis.org

1, 1, 1, 2, 8, 29, 86, 224, 554, 1381, 3556, 9382, 24901, 65737, 172321, 450017, 1174985, 3072365, 8044478, 21074012, 55199573, 144535714, 378366976, 990441502, 2592800365, 6787973872, 17771619370, 46527959417, 121813193825, 318910531073, 834913179137
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A000930, A005251, A024493, A099099, A369845.
Cf. A107025, A373904, A373906.

Programs

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

Formula

a(n) = 6*a(n-1) - 15*a(n-2) + 21*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: 1/(1 - x - x^3/(1 - x)^5).

A373906 a(n) = Sum_{k=0..floor(n/4)} binomial(n+2*k,n-4*k).

Original entry on oeis.org

1, 1, 1, 1, 2, 8, 29, 85, 212, 476, 1016, 2172, 4825, 11213, 26763, 64095, 151851, 354737, 820328, 1889968, 4361521, 10106859, 23509678, 54793282, 127709888, 297336790, 691382201, 1606284377, 3731020629, 8668253125, 20146856893, 46840732201, 108918637566, 253262275888
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A003269, A003522, A005252, A038503, A099131.
Cf. A107025, A373904, A373905.

Programs

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

Formula

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

A373961 Number of compositions of 6*n-1 into parts 5 and 6.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 15, 44, 129, 340, 804, 1742, 3550, 7009, 13835, 28033, 58993, 128136, 282569, 622575, 1357136, 2918449, 6204578, 13104675, 27646776, 58502733, 124411595, 265807567, 569552644, 1221316021, 2616456236, 5595314908, 11944318042, 25466629978
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A107025, A369794, A373962, A373963, A373964.
Cf. A017837.

Programs

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

Formula

a(n) = A017837(6*n-1).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+k,n-1-5*k).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 7*a(n-5) - a(n-6).
G.f.: x*(1-x)^4/((1-x)^6 - x^5).
a(n) = A373962(n+1) - A373962(n).

A373962 Number of compositions of 6*n-2 into parts 5 and 6.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 22, 37, 81, 210, 550, 1354, 3096, 6646, 13655, 27490, 55523, 114516, 242652, 525221, 1147796, 2504932, 5423381, 11627959, 24732634, 52379410, 110882143, 235293738, 501101305, 1070653949, 2291969970, 4908426206, 10503741114, 22448059156
Offset: 1

Views

Author

Seiichi Manyama, Jun 24 2024

Keywords

Links

Crossrefs

Cf. A107025, A369794, A373961, A373963, A373964.
Cf. A017837.

Programs

  • Mathematica
    LinearRecurrence[{6,-15,20,-15,7,-1},{0,1,3,6,10,15},40] (* Harvey P. Dale, Nov 16 2024 *)
  • PARI
    a(n) = sum(k=0, n\5, binomial(n+k, n-2-5*k));

Formula

a(n) = A017837(6*n-2).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+k,n-2-5*k).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 7*a(n-5) - a(n-6).
G.f.: x^2*(1-x)^3/((1-x)^6 - x^5).
a(n) = A373963(n+1) - A373963(n).

A373963 Number of compositions of 6*n-3 into parts 5 and 6.

Original entry on oeis.org

0, 0, 1, 4, 10, 20, 35, 57, 94, 175, 385, 935, 2289, 5385, 12031, 25686, 53176, 108699, 223215, 465867, 991088, 2138884, 4643816, 10067197, 21695156, 46427790, 98807200, 209689343, 444983081, 946084386, 2016738335, 4308708305, 9217134511, 19720875625
Offset: 1

Views

Author

Seiichi Manyama, Jun 24 2024

Keywords

Links

Crossrefs

Cf. A107025, A369794, A373961, A373962, A373964.
Cf. A017837.

Programs

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

Formula

a(n) = A017837(6*n-3).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+k,n-3-5*k).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 7*a(n-5) - a(n-6).
G.f.: x^3*(1-x)^2/((1-x)^6 - x^5).
a(n) = A373964(n+1) - A373964(n).

A373964 Number of compositions of 6*n-4 into parts 5 and 6.

Original entry on oeis.org

0, 0, 0, 1, 5, 15, 35, 70, 127, 221, 396, 781, 1716, 4005, 9390, 21421, 47107, 100283, 208982, 432197, 898064, 1889152, 4028036, 8671852, 18739049, 40434205, 86861995, 185669195, 395358538, 840341619, 1786426005, 3803164340, 8111872645, 17329007156
Offset: 1

Views

Author

Seiichi Manyama, Jun 24 2024

Keywords

Links

Crossrefs

Cf. A107025, A369794, A373961, A373962, A373963.
Cf. A017837.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n+k, n-4-5*k));

Formula

a(n) = A017837(6*n-4).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+k,n-4-5*k).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 7*a(n-5) - a(n-6).
G.f.: x^4*(1-x)/((1-x)^6 - x^5).
a(n) = A369794(n+1) - A369794(n).
Showing 1-10 of 11 results. Next

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