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.

User: Seiichi Manyama

Seiichi Manyama's wiki page.

Seiichi Manyama has authored 9293 sequences. Here are the ten most recent ones:

A387652 a(n) = Sum_{k=0..floor(n/3)} 2^(n-3*k) * binomial(2*k+1,2*n-6*k+1).

Original entry on oeis.org

1, 0, 0, 3, 2, 0, 5, 20, 4, 7, 70, 84, 17, 168, 504, 299, 346, 1848, 2653, 1452, 5180, 13743, 12350, 14508, 51561, 81440, 68432, 162323, 391026, 442544, 555445, 1498116, 2500276, 2666711, 5202550, 11465284, 14985153, 20025432, 45011176, 77173371, 95454666, 168802152
Offset: 0

Views

Author

Seiichi Manyama, Sep 05 2025

Keywords

Links

Crossrefs

Cf. A387627, A387651.
Cf. A387648.

Programs

  • PARI
    a(n) = sum(k=0, n\3, 2^(n-3*k)*binomial(2*k+1, 2*n-6*k+1));

Formula

G.f.: (1+x^3-2*x^4)/((1+x^3-2*x^4)^2 - 4*x^3).
a(n) = 2*a(n-3) + 4*a(n-4) - a(n-6) + 4*a(n-7) - 4*a(n-8).

A387626 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(2*n-6*k+1,2*k).

Original entry on oeis.org

1, 1, 1, 1, 7, 21, 43, 73, 131, 297, 715, 1593, 3259, 6553, 13723, 29833, 64827, 137881, 289179, 608329, 1293083, 2762457, 5885179, 12478601, 26418363, 56028761, 119072987, 253139017, 537620571, 1140840793, 2420927291, 5139947401, 10916332411, 23182447833
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2025

Keywords

Links

Crossrefs

Cf. A002315, A387624, A387625.

Programs

  • Magma
    [&+[2^k* Binomial(2*n-6*k+1, 2*k): k in [0..Floor (n/4)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025
  • Mathematica
    Table[Sum[2^k*Binomial[2*n-6*k+1,2*k],{k,0,Floor[n/4]}],{n,0,40}] (* Vincenzo Librandi, Sep 04 2025 *)
  • PARI
    a(n) = sum(k=0, n\4, 2^k*binomial(2*n-6*k+1, 2*k));

Formula

G.f.: (1-x+2*x^4)/((1-x+2*x^4)^2 - 8*x^4).
a(n) = 2*a(n-1) - a(n-2) + 4*a(n-4) + 4*a(n-5) - 4*a(n-8).

A387625 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(2*n-4*k+1,2*k).

Original entry on oeis.org

1, 1, 1, 7, 21, 43, 93, 251, 661, 1587, 3805, 9499, 23813, 58691, 144141, 356491, 883637, 2184115, 5391869, 13325371, 32953317, 81459235, 201299565, 497518187, 1229819541, 3039854611, 7513347421, 18570354203, 45900859333, 113454099843, 280422868685
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2025

Keywords

Links

Crossrefs

Cf. A002315, A387624, A387626.

Programs

  • PARI
    a(n) = sum(k=0, n\3, 2^k*binomial(2*n-4*k+1, 2*k));

Formula

G.f.: (1-x+2*x^3)/((1-x+2*x^3)^2 - 8*x^3).
a(n) = 2*a(n-1) - a(n-2) + 4*a(n-3) + 4*a(n-4) - 4*a(n-6).

A387624 a(n) = Sum_{k=0..floor(n/2)} 2^k * binomial(2*n-2*k+1,2*k).

Original entry on oeis.org

1, 1, 7, 21, 63, 213, 671, 2149, 6911, 22101, 70847, 227045, 727391, 2330901, 7468767, 23931621, 76683583, 245713493, 787329151, 2522806629, 8083720351, 25902323221, 82997717407, 265946059365, 852159682431, 2730539119701, 8749350654527, 28035173160421
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2025

Keywords

Links

Crossrefs

Cf. A002315, A387625, A387626.
Cf. A387627.

Programs

  • PARI
    a(n) = sum(k=0, n\2, 2^k*binomial(2*n-2*k+1, 2*k));

Formula

G.f.: (1-x+2*x^2)/((1-x+2*x^2)^2 - 8*x^2).
a(n) = 2*a(n-1) + 3*a(n-2) + 4*a(n-3) - 4*a(n-4).

A387623 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(2*n-6*k,2*k).

Original entry on oeis.org

1, 1, 1, 1, 3, 13, 31, 57, 95, 193, 463, 1081, 2295, 4609, 9423, 20185, 44071, 94801, 199807, 418921, 885879, 1889889, 4034639, 8573561, 18155399, 38461105, 81665695, 173627401, 368961431, 783201921, 1661811055, 3527298329, 7490519335, 15908549329, 33779968447
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2025

Keywords

Links

Crossrefs

Cf. A001541, A108480, A387622.
Cf. A387626, A387629.

Programs

  • PARI
    a(n) = sum(k=0, n\4, 2^k*binomial(2*n-6*k, 2*k));

Formula

G.f.: (1-x-2*x^4)/((1-x-2*x^4)^2 - 8*x^5).
a(n) = 2*a(n-1) - a(n-2) + 4*a(n-4) + 4*a(n-5) - 4*a(n-8).

A387622 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(2*n-4*k,2*k).

Original entry on oeis.org

1, 1, 1, 3, 13, 31, 61, 151, 413, 1031, 2445, 5991, 15069, 37447, 91917, 226503, 561373, 1389735, 3431501, 8474983, 20955229, 51814407, 128054029, 316455559, 782209629, 1933537511, 4779082829, 11812031271, 29195752157, 72164132167, 178368130061
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2025

Keywords

Links

Crossrefs

Cf. A001541, A108480, A387623.

Programs

  • PARI
    a(n) = sum(k=0, n\3, 2^k*binomial(2*n-4*k, 2*k));

Formula

G.f.: (1-x-2*x^3)/((1-x-2*x^3)^2 - 8*x^4).
a(n) = 2*a(n-1) - a(n-2) + 4*a(n-3) + 4*a(n-4) - 4*a(n-6).

A387627 a(n) = Sum_{k=0..floor(n/2)} 2^k * binomial(2*n-2*k+1,2*k+1).

Original entry on oeis.org

1, 3, 7, 27, 83, 263, 855, 2723, 8731, 27999, 89663, 287355, 920771, 2950263, 9453607, 30291667, 97062123, 311012623, 996563855, 3193247403, 10231988371, 32785923879, 105054547063, 336621829635, 1078623042491, 3456186066623, 11074510391007, 35485583833307
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2025

Keywords

Links

Crossrefs

Cf. A001653, A387628, A387629.
Cf. A099511.

Programs

  • Magma
    [&+[2^k* Binomial(2*n-2*k+1, 2*k+1): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025
  • Mathematica
    Table[Sum[2^k*Binomial[2*n-2*k+1,2*k+1],{k,0,Floor[n/2]}],{n,0,40}] (* Vincenzo Librandi, Sep 04 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, 2^k*binomial(2*n-2*k+1, 2*k+1));

Formula

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

A387628 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(2*n-4*k+1,2*k+1).

Original entry on oeis.org

1, 3, 5, 9, 29, 81, 185, 429, 1093, 2785, 6817, 16613, 41181, 102441, 253049, 623693, 1541557, 3814929, 9430545, 23297397, 57577997, 142345721, 351858985, 869614109, 2149341925, 5312698977, 13131636417, 32457015109, 80223121469, 198288112969, 490110342873
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2025

Keywords

Links

Crossrefs

Cf. A001653, A387627, A387629.

Programs

  • Magma
    [&+[2^k* Binomial(2*n-4*k+1, 2*k+1): k in [0..Floor (n/3)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025
  • Mathematica
    Table[Sum[2^k*Binomial[2*n-4*k+1,2*k+1],{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 04 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, 2^k*binomial(2*n-4*k+1, 2*k+1));

Formula

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

A387629 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(2*n-6*k+1,2*k+1).

Original entry on oeis.org

1, 3, 5, 7, 11, 31, 83, 183, 351, 675, 1435, 3231, 7119, 14987, 30963, 64871, 138775, 298403, 636091, 1344191, 2838399, 6021371, 12818467, 27277207, 57911207, 122790675, 260485131, 553185519, 1175285967, 2496108459, 5298760307, 11246985927, 23877452663, 50702334403
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2025

Keywords

Links

Crossrefs

Cf. A001653, A387627, A387628.
Cf. A387623, A387626.

Programs

  • Magma
    [&+[2^k* Binomial(2*n-6*k+1, 2*k+1): k in [0..Floor (n/4)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025
  • Mathematica
    Table[Sum[2^k*Binomial[2*n-6*k+1,2*k+1],{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 04 2025 *)
  • PARI
    a(n) = sum(k=0, n\4, 2^k*binomial(2*n-6*k+1, 2*k+1));

Formula

G.f.: (1+x-2*x^4)/((1+x-2*x^4)^2 - 4*x).
a(n) = 2*a(n-1) - a(n-2) + 4*a(n-4) + 4*a(n-5) - 4*a(n-8).

A387601 a(n) = (1/2) * Sum_{k=0..floor(n/4)} 2^k * binomial(2*n-6*k+2,2*k+1).

Original entry on oeis.org

1, 2, 3, 4, 9, 26, 63, 128, 241, 486, 1075, 2412, 5189, 10770, 22343, 47352, 101801, 218142, 462635, 976260, 2065741, 4391914, 9351823, 19877904, 42164785, 89409718, 189779059, 403162268, 856453269, 1818474626, 3859843799, 8193466664, 17396892537, 36942391118
Offset: 0

Views

Author

Seiichi Manyama, Sep 02 2025

Keywords

Links

Crossrefs

Cf. A387508.

Programs

  • Magma
    [&+[2^k* Binomial(2*n-6*k+2, 2*k+1)/2: k in [0..Floor (n/4)]]: n in [0..35]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[2^k*Binomial[2*n-6*k+2, 2*k+1]/2,{k,0,Floor[n/4]}],{n,0,40}] (* Vincenzo Librandi, Sep 03 2025 *)
  • PARI
    a(n) = sum(k=0, n\4, 2^k*binomial(2*n-6*k+2, 2*k+1))/2;

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A387508.
G.f.: 1/((1-x-2*x^4)^2 - 8*x^5).
a(n) = 2*a(n-1) - a(n-2) + 4*a(n-4) + 4*a(n-5) - 4*a(n-8).

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

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