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 1-9 of 9 results.

A100136 a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 3^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 13, 31, 61, 106, 169, 262, 424, 748, 1417, 2749, 5251, 9709, 17395, 30553, 53434, 94285, 168859, 306283, 558742, 1017895, 1844044, 3320044, 5952472, 10660177, 19119385, 34383781, 61985497, 111884665, 201938701, 364128136
Offset: 0

Views

Author

Paul Barry, Nov 06 2004

Keywords

Links

Crossrefs

Cf. A098576, A100134, A100135.

Formula

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

A100137 a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 2^(n-6k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 65, 136, 296, 672, 1584, 3840, 9473, 23566, 58736, 146080, 361760, 891328, 2184961, 5331476, 12958684, 31400160, 75910320, 183220800, 441787201, 1064687642, 2565404524, 6181873208, 14899796416, 35922756992, 86635757825
Offset: 0

Views

Author

Paul Barry, Nov 06 2004

Keywords

Comments

Binomial transform of 1,1,1,1,1,1,2,2,2,5,5,11,11,... with g.f. (1-x)^2(1+x)^2/(1-3x^2+3x^4-2x^6)=(1+x)(1-x^2)^2/((1-x^2)^3-x^6).

Links

Crossrefs

Cf. A024493, A100131, A100134, A100137, A100138.

Programs

  • Mathematica
    Table[Sum[Binomial[n-3k,3k]2^(n-6k),{k,0,Floor[n/6]}],{n,0,30}] (* or *) LinearRecurrence[{6,-12,8,0,0,1},{1,2,4,8,16,32},31] (* Harvey P. Dale, Mar 19 2015 *)

Formula

G.f.: (1-2x)^2/((1-2x)^3 - x^6).
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) + a(n-6).

A100138 a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 2^(n-5k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 66, 144, 336, 832, 2144, 5632, 14852, 38968, 101312, 260736, 664704, 1681152, 4226056, 10578080, 26407648, 65838848, 164095360, 409129472, 1020795408, 2549137824, 6371133120, 15935185792, 39878810624, 99837958144
Offset: 0

Views

Author

Paul Barry, Nov 06 2004

Keywords

Comments

Binomial transform of 1,1,1,1,1,1,3,3,9,9,21,... with g.f. (1-x)^2(1+x)^2/(1-3x^2+3x^4-3x^6)=(1+x)(1-x^2)^2/((1-x^2)^3-2x^6).

Links

Crossrefs

Cf. A052101, A100132, A100134, A100137, A100139.

Programs

  • Mathematica
    Table[Sum[Binomial[n-3k,3k]2^(n-5k),{k,0,Floor[n/6]}],{n,0,30}] (* or *) LinearRecurrence[{6,-12,8,0,0,2},{1,2,4,8,16,32},30] (* Harvey P. Dale, Dec 30 2019 *)

Formula

G.f.: (1-2x)^2/((1-2x)^3 - 2x^6).
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) + 2*a(n-6).

A348289 a(n) = Sum_{k=0..floor(n/8)} binomial(n-4*k,4*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 6, 16, 36, 71, 127, 211, 331, 497, 725, 1047, 1531, 2316, 3668, 6064, 10312, 17717, 30309, 51165, 84893, 138417, 222329, 353285, 558253, 881918, 1399274, 2236480, 3604588, 5853067, 9553715, 15631615, 25570103, 41734433, 67889133, 110035211, 177778263
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2021

Keywords

Links

Crossrefs

Cf. A000045, A005252, A100134, A348290.
Cf. A101552.

Programs

  • PARI
    a(n) = sum(k=0, n\8, binomial(n-4*k, 4*k));
  • PARI
    my(N=66, x='x+O('x^N)); Vec((1-x)^3/((1-x)^4-x^8))

Formula

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

A100135 a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 2^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 9, 21, 41, 71, 113, 173, 269, 443, 777, 1413, 2577, 4615, 8065, 13813, 23413, 39691, 67801, 116973, 203337, 354519, 617345, 1071197, 1851677, 3192731, 5501033, 9485621, 16381185, 28330119, 49035777, 84883621, 146875717, 253983307, 438968761
Offset: 0

Views

Author

Paul Barry, Nov 06 2004

Keywords

Links

Crossrefs

Cf. A098575, A100134, A100136.

Programs

  • Mathematica
    LinearRecurrence[{3,-3, 1, 0, 0, 2},{1,1,1,1,1,1},38] (* James C. McMahon, Dec 22 2023 *)

Formula

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

A100139 a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 3^k * 2^(n-6k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 67, 152, 376, 992, 2704, 7424, 20233, 54398, 144112, 376736, 974368, 2500544, 6385435, 16264220, 41396788, 105423776, 268818064, 686499008, 1755723793, 4495691834, 11521647916, 29543647160, 75774096832, 194353495424
Offset: 0

Views

Author

Paul Barry, Nov 06 2004

Keywords

Comments

Binomial transform of 1,1,1,1,1,1,4,4,13,13,31,... with g.f. (1-x)^2(1+x)^3/(1-3x^2+3x^4-4x^6)=(1+x)(1-x^2)^2/((1-x^2)^3-3x^6).

Links

Crossrefs

Cf. A097122, A100133, A100134, A100137, A100138.

Programs

  • Mathematica
    LinearRecurrence[{6,-12,8,0,0,3},{1,2,4,8,16,32},30] (* Harvey P. Dale, Sep 30 2015 *)

Formula

G.f.: (1-2x)^2/((1-2x)^3 - 3x^6).
a(n) = 6*a(n-1) -12*a(n-2) + 8*a(n-3) + 3*a(n-6).

A348290 a(n) = Sum_{k=0..floor(n/10)} binomial(n-5*k,5*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 253, 463, 793, 1288, 2003, 3005, 4380, 6255, 8855, 12630, 18508, 28358, 45783, 77408, 134883, 237888, 418513, 727513, 1243163, 2083888, 3426771, 5535911, 8808206, 13850761, 21615771, 33638409, 52455339, 82332229, 130506914, 209273284
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2021

Keywords

Links

Crossrefs

Cf. A000045, A005252, A100134, A348289.

Programs

  • PARI
    a(n) = sum(k=0, n\10, binomial(n-5*k, 5*k));
  • PARI
    my(N=66, x='x+O('x^N)); Vec((1-x)^4/((1-x)^5-x^10))

Formula

G.f.: (1-x)^4/((1-x)^5 - x^10).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + a(n-10).

A348308 a(n) = Sum_{k=0..floor(n/6)} (-1)^k * binomial(n-3*k,3*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, -3, -9, -19, -34, -55, -82, -112, -136, -135, -75, 99, 469, 1147, 2269, 3970, 6325, 9235, 12231, 14166, 12771, 4076, -18244, -63424, -143695, -273223, -464779, -722439, -1027959, -1317915, -1448612, -1146827, 52219, 2870965, 8337370, 17769349, 32615514, 54022692, 81938664
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2021

Keywords

Links

Crossrefs

Cf. A348309, A348310.
Cf. A057681, A100134.

Programs

  • Mathematica
    LinearRecurrence[{3, -3, 1, 0, 0, -1}, {1, 1, 1, 1, 1, 1}, 45] (* Amiram Eldar, Oct 11 2021 *)
  • PARI
    a(n) = sum(k=0, n\6, (-1)^k*binomial(n-3*k, 3*k));
  • PARI
    my(N=66, x='x+O('x^N)); Vec((1-x)^2/((1-x)^3+x^6))

Formula

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

A370722 a(n) = Sum_{k=0..floor(n/7)} binomial(n-4*k,3*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 5, 11, 21, 36, 57, 85, 122, 173, 249, 371, 575, 918, 1485, 2398, 3830, 6030, 9369, 14422, 22107, 33909, 52226, 80888, 125925, 196706, 307653, 480873, 750275, 1168085, 1815191, 2817518, 4371772, 6785606, 10539893, 16384908, 25488736
Offset: 0

Views

Author

Seiichi Manyama, Feb 28 2024

Keywords

Links

Crossrefs

Cf. A003522, A100134, A137356.
Cf. A003520, A005689, A348289.

Programs

  • Mathematica
    LinearRecurrence[{3, -3, 1, 0, 0, 0, 1}, Table[1, 7], 50] (* Paolo Xausa, Mar 15 2024 *)
  • PARI
    a(n) = sum(k=0, n\7, binomial(n-4*k, 3*k));
  • PARI
    my(N=50, x='x+O('x^N)); Vec((1-x)^2/((1-x)^3-x^7))

Formula

G.f.: (1-x)^2/((1-x)^3 - x^7).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-7).
Showing 1-9 of 9 results.

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