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 9278 sequences. Here are the ten most recent ones:

A387557 a(n) = (1/2) * Sum_{k=0..floor(n/3)} 2^(n-3*k) * binomial(2*k+2,2*n-6*k+1).

Original entry on oeis.org

1, 0, 0, 2, 4, 0, 3, 20, 12, 4, 56, 112, 37, 120, 504, 486, 300, 1584, 3175, 2124, 4196, 13736, 16576, 14560, 46217, 92336, 87024, 145226, 391124, 540192, 584267, 1397444, 2742332, 3162828, 4973640, 11517840, 17306989, 21377448, 43440616, 82902062, 108691196
Offset: 0

Views

Author

Seiichi Manyama, Sep 02 2025

Keywords

Links

Crossrefs

Cf. A387516.

Programs

  • Magma
    [&+[2^(n-3*k) * Binomial(2*k+2, 2*n-6*k+1)/2: k in [0..Floor(n/3)]]: n in [0..40]]; // Vincenzo Librandi, Sep 02 2025
  • Mathematica
    Table[Sum[2^(n-3*k)*Binomial[2*k+2, 2*n-6*k+1]/2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 02 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, 2^(n-3*k)*binomial(2*k+2, 2*n-6*k+1))/2;

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A387516.
G.f.: 1/((1-x^3-2*x^4)^2 - 8*x^7).
a(n) = 2*a(n-3) + 4*a(n-4) - a(n-6) + 4*a(n-7) - 4*a(n-8).

A387556 a(n) = (1/2) * Sum_{k=0..floor(n/3)} 2^(n-2*k) * binomial(2*k+2,2*n-6*k+1).

Original entry on oeis.org

1, 0, 0, 4, 8, 0, 12, 80, 48, 32, 448, 896, 336, 1920, 8064, 7872, 8320, 50688, 101824, 79616, 262400, 879616, 1096704, 1490944, 5888256, 11923456, 13332480, 34886656, 100288512, 146227200, 228961280, 702910464, 1430450176, 1968660480, 4587044864
Offset: 0

Views

Author

Seiichi Manyama, Sep 02 2025

Keywords

Links

Crossrefs

Cf. A387485.

Programs

  • Magma
    [&+[2^(n-2*k) * Binomial(2*k+2, 2*n-6*k+1)/2: k in [0..Floor(n/3)]]: n in [0..40]]; // Vincenzo Librandi, Sep 02 2025
  • Mathematica
    Table[Sum[2^(n-2*k)*Binomial[2*k+2, 2*n-6*k+1]/2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 02 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, 2^(n-2*k)*binomial(2*k+2, 2*n-6*k+1))/2;

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A387485.
G.f.: 1/((1-2*x^3-4*x^4)^2 - 32*x^7).
a(n) = 4*a(n-3) + 8*a(n-4) - 4*a(n-6) + 16*a(n-7) - 16*a(n-8).

A387555 a(n) = (1/2) * Sum_{k=0..floor(n/3)} 2^(n-k) * binomial(2*k+2,2*n-6*k+1).

Original entry on oeis.org

1, 0, 0, 8, 16, 0, 48, 320, 192, 256, 3584, 7168, 3328, 30720, 129024, 129024, 245760, 1622016, 3272704, 3293184, 16596992, 56360960, 74776576, 166985728, 752156672, 1552941056, 2268069888, 8638693376, 25806503936, 41498443776, 99265544192, 357275009024
Offset: 0

Views

Author

Seiichi Manyama, Sep 02 2025

Keywords

Links

Crossrefs

Cf. A387484.

Programs

  • Magma
    [&+[2^(n-k) * Binomial(2*k+2, 2*n-6*k+1)/2: k in [0..Floor(n/3)]]: n in [0..40]]; // Vincenzo Librandi, Sep 02 2025
  • Mathematica
    Table[Sum[2^(n-k)*Binomial[2*k+2, 2*n-6*k+1]/2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 02 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, 2^(n-k)*binomial(2*k+2, 2*n-6*k+1))/2;

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A387484.
G.f.: 1/((1-4*x^3-8*x^4)^2 - 128*x^7).
a(n) = 8*a(n-3) + 16*a(n-4) - 16*a(n-6) + 64*a(n-7) - 64*a(n-8).

A387554 a(n) = (1/2) * Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(2*k+2,2*n-4*k+1).

Original entry on oeis.org

1, 0, 2, 4, 3, 20, 16, 56, 117, 152, 510, 700, 1671, 3532, 5772, 14480, 24761, 52400, 109114, 198324, 437899, 821828, 1670536, 3423784, 6547325, 13666184, 26654966, 53492716, 108440335, 212433276, 432672004, 857090304, 1713987777, 3452225824, 6839636530
Offset: 0

Views

Author

Seiichi Manyama, Sep 02 2025

Keywords

Links

Crossrefs

Cf. A123957, A387515.

Programs

  • Magma
    [&+[2^(n-2*k) * Binomial(2*k+2, 2*n-4*k+1)/2: k in [0..Floor(n/2)]]: n in [0..40]]; // Vincenzo Librandi, Sep 02 2025
  • Mathematica
    Table[Sum[2^(n-2*k)*Binomial[2*k+2, 2*n-4*k+1]/2,{k,0,Floor[n/2]}],{n,0,40}] (* Vincenzo Librandi, Sep 02 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, 2^(n-2*k)*binomial(2*k+2, 2*n-4*k+1))/2;

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A387515.
G.f.: 1/((1-x^2-2*x^3)^2 - 8*x^5).
a(n) = 2*a(n-2) + 4*a(n-3) - a(n-4) + 4*a(n-5) - 4*a(n-6).

A387553 a(n) = (1/2) * Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(2*k+2,2*n-4*k+1).

Original entry on oeis.org

1, 0, 4, 8, 12, 80, 80, 448, 976, 2176, 8256, 14720, 52416, 124672, 313600, 956416, 2145536, 6438912, 16135168, 42117120, 117754880, 290820096, 812109824, 2091991040, 5519691776, 14911766528, 38335299584, 103777271808, 271034662912, 716987629568, 1911288823808
Offset: 0

Views

Author

Seiichi Manyama, Sep 02 2025

Keywords

Links

Crossrefs

Cf. A387483.

Programs

  • Magma
    [&+[2^(n-k) * Binomial(2*k+2, 2*n-4*k+1)/2: k in [0..Floor(n/2)]]: n in [0..40]]; // Vincenzo Librandi, Sep 02 2025
  • Mathematica
    Table[Sum[2^(n-k)*Binomial[2*k+2, 2*n-4*k+1]/2,{k,0,Floor[n/2]}],{n,0,40}] (* Vincenzo Librandi, Sep 02 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, 2^(n-k)*binomial(2*k+2, 2*n-4*k+1))/2;

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A387483.
G.f.: 1/((1-2*x^2-4*x^3)^2 - 32*x^5).
a(n) = 4*a(n-2) + 8*a(n-3) - 4*a(n-4) + 16*a(n-5) - 16*a(n-6).

A387552 a(n) = (1/2) * Sum_{k=0..floor(n/3)} 2^k * binomial(2*k+2,2*n-6*k+1).

Original entry on oeis.org

1, 0, 0, 4, 4, 0, 12, 40, 12, 32, 224, 224, 112, 960, 2016, 1152, 3600, 12672, 13120, 15168, 64256, 110848, 99904, 291200, 734912, 836608, 1376256, 4114432, 6516224, 8042496, 20953088, 43890688, 56483072, 107188224, 260720640, 404997120, 609147904, 1431527424
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2025

Keywords

Links

Crossrefs

Cf. A387477.

Programs

  • Magma
    [&+[2^k * Binomial(2*k+2, 2*n-6*k+1)/2: k in [0..Floor(n/3)]]: n in [0..40]]; // Vincenzo Librandi, Sep 02 2025
  • Mathematica
    Table[Sum[2^k*Binomial[2*k+2, 2*n-6*k+1]/2,{k,0,Floor[n/3]}],{n,0,40}] (Vincenzo Librandi, Sep 02 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, 2^k*binomial(2*k+2, 2*n-6*k+1))/2;

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A387477.
G.f.: 1/((1-2*x^3-2*x^4)^2 - 16*x^7).
a(n) = 4*a(n-3) + 4*a(n-4) - 4*a(n-6) + 8*a(n-7) - 4*a(n-8).

A387551 a(n) = (1/2) * Sum_{k=0..floor(n/2)} 2^k * binomial(2*k+2,2*n-4*k+1).

Original entry on oeis.org

1, 0, 4, 4, 12, 40, 44, 224, 304, 992, 2208, 4480, 13200, 24320, 68608, 145856, 345920, 848256, 1834432, 4644864, 10239488, 24708096, 57602048, 132493312, 318103808, 724885504, 1728687104, 4003968000, 9371413504, 22045935616, 51113446400, 120583479296
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2025

Keywords

Links

Crossrefs

Cf. A387476.

Programs

  • Magma
    [&+[2^k * Binomial(2*k+2, 2*n-4*k+1)/2: k in [0..Floor(n/2)]]: n in [0..40]]; // Vincenzo Librandi, Sep 02 2025
  • Mathematica
    Table[Sum[2^k*Binomial[2*k+2, 2*n-4*k+1]/2,{k,0,Floor[n/2]}],{n,0,40}] (* Vincenzo Librandi, Sep 02 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, 2^k*binomial(2*k+2, 2*n-4*k+1))/2;

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A387476.
G.f.: 1/((1-2*x^2-2*x^3)^2 - 16*x^5).
a(n) = 4*a(n-2) + 4*a(n-3) - 4*a(n-4) + 8*a(n-5) - 4*a(n-6).

A387550 a(n) = (1/2) * Sum_{k=0..n} 2^k * binomial(2*k+2,2*n-2*k+1).

Original entry on oeis.org

1, 4, 16, 72, 316, 1376, 6016, 26304, 114960, 502464, 2196224, 9599360, 41957312, 183389184, 801566720, 3503527936, 15313395968, 66932560896, 292552200192, 1278701856768, 5589014330368, 24428744679424, 106774384771072, 466694846300160, 2039853285314560
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2025

Keywords

Links

Crossrefs

Cf. A375276.

Programs

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

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A375276.
G.f.: 1/((1-2*x-2*x^2)^2 - 16*x^3).
a(n) = 4*a(n-1) + 8*a(n-3) - 4*a(n-4).

A387516 a(n) = Sum_{k=0..floor(n/3)} 2^(n-3*k) * binomial(k,n-3*k)^2.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 1, 8, 4, 1, 18, 36, 9, 32, 144, 129, 66, 400, 801, 472, 932, 3201, 3698, 2916, 9865, 19728, 17248, 28225, 78690, 105536, 106625, 262408, 516388, 566785, 871730, 2064964, 3040713, 3585888, 7366032, 14098817, 17860962, 27066384, 56844833, 88593688
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2025

Keywords

Links

Crossrefs

Cf. A387484, A387485.
Cf. A387482.

Programs

  • Magma
    [(&+[2^(n-3*k) * Binomial(k,n-3*k)^2: k in [0..Floor(n/2)]]): n in [0..40]]; // Vincenzo Librandi, Sep 01 2025
  • Mathematica
    Table[Sum[2^(n-3*k)*Binomial[k,n-3*k]^2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 01 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, 2^(n-3*k)*binomial(k, n-3*k)^2);

Formula

G.f.: 1/sqrt((1-x^3-2*x^4)^2 - 8*x^7).

A387515 a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(k,n-2*k)^2.

Original entry on oeis.org

1, 0, 1, 2, 1, 8, 5, 18, 37, 40, 145, 178, 417, 872, 1301, 3330, 5365, 11080, 22801, 39362, 86721, 157128, 312293, 631666, 1169541, 2416104, 4602961, 9061458, 18123553, 34717608, 69825013, 135902818, 267384405, 531611656, 1035512785, 2060791650, 4048647489, 7979180296
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2025

Keywords

Links

Crossrefs

Cf. A108488, A387516.
Cf. A387481, A387483.

Programs

  • Magma
    [(&+[2^(n-2*k) * Binomial(k,n-2*k)^2: k in [0..Floor(n/2)]]): n in [0..40]]; // Vincenzo Librandi, Sep 01 2025
  • Mathematica
    Table[Sum[2^(n-2*k)*Binomial[k,n-2*k]^2,{k,0,Floor[n/2]}],{n,0,40}] (* Vincenzo Librandi, Sep 01 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, 2^(n-2*k)*binomial(k, n-2*k)^2);

Formula

G.f.: 1/sqrt((1-x^2-2*x^3)^2 - 8*x^5).

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