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 25 results. Next

A011842 a(n) = floor(n*(n-1)*(n-2)/24).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 8, 14, 21, 30, 41, 55, 71, 91, 113, 140, 170, 204, 242, 285, 332, 385, 442, 506, 575, 650, 731, 819, 913, 1015, 1123, 1240, 1364, 1496, 1636, 1785, 1942, 2109, 2284, 2470, 2665, 2870, 3085, 3311, 3547, 3795, 4053, 4324, 4606, 4900, 5206, 5525, 5856, 6201, 6558, 6930, 7315, 7714, 8127, 8555, 8997, 9455, 9927, 10416, 10920
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

A column of triangle A011847.
Cf. A011886.

Programs

  • Magma
    [Floor(Binomial(n,3)/4): n in [0..80]]; // G. C. Greubel, Oct 20 2024
  • Maple
    seq(floor(binomial(n,3)/4), n=0..43); # Zerinvary Lajos, Jan 12 2009
  • Mathematica
    Floor[Binomial[Range[0,80], 3]/4] (* G. C. Greubel, Oct 20 2024 *)
  • SageMath
    [binomial(n,3)//4 for n in range(81)] # G. C. Greubel, Oct 20 2024

Formula

From R. J. Mathar, Apr 15 2010: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-8) - 3*a(n-9) + 3*a(n-10) - a(n-11).
G.f.: x^4*(1-x+x^2)*(1+x^2-x^3+x^4) / ((1-x)^4*(1+x)*(1+x^2)*(1+x^4)). (End)
a(n) = floor(binomial(n+1,4)/(n+1)). - Gary Detlefs, Nov 23 2011

Extensions

More terms added by G. C. Greubel, Oct 20 2024

A011896 a(n) = floor( n*(n-1)*(n-2)/14 ).

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 8, 15, 24, 36, 51, 70, 94, 122, 156, 195, 240, 291, 349, 415, 488, 570, 660, 759, 867, 985, 1114, 1253, 1404, 1566, 1740, 1926, 2125, 2338, 2564, 2805, 3060, 3330, 3615, 3916, 4234, 4568, 4920, 5289, 5676, 6081, 6505, 6949, 7412, 7896
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886, A055610.

Programs

  • Magma
    [Floor(3*Binomial(n,3)/7): n in [0..50]]; // G. C. Greubel, Oct 16 2024
  • Mathematica
    Table[Floor[(n(n-1)(n-2))/14],{n,0,50}] (* or  *)
LinearRecurrence[{3,-3,1,0,0,0,1,-3,3,-1},{0,0,0,0,1,4,8,15,24,36},50] (* Harvey P. Dale, Jan 03 2024 *)
  • PARI
    a(n)=n*(n-1)*(n-2)\14
  • SageMath
    [3*binomial(n,3)//7 for n in range(51)] # G. C. Greubel, Oct 16 2024

Formula

G.f.: x^4*(1+x-x^2+2*x^3-x^4+x^5)/((1-x)^3*(1-x^7)).
a(2-n) = (-1)*A055610(n).

Extensions

Additional comments from Michael Somos, Jun 02 2000.

A011887 a(n) = floor( n*(n-1)*(n-2)/5 ).

Original entry on oeis.org

0, 0, 0, 1, 4, 12, 24, 42, 67, 100, 144, 198, 264, 343, 436, 546, 672, 816, 979, 1162, 1368, 1596, 1848, 2125, 2428, 2760, 3120, 3510, 3931, 4384, 4872, 5394, 5952, 6547, 7180, 7854, 8568, 9324, 10123, 10966
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(n*(n-1)*(n-2)/5): n in [0..50]]; // Vincenzo Librandi, Jul 07 2012
  • Mathematica
    CoefficientList[Series[x^3*(1+x+3*x^2-x^3+2*x^4)/((1-x)^3*(1-x^5)),{x,0,50}] ,x] (* Vincenzo Librandi, Jul 07 2012 *)
  • SageMath
    [6*binomial(n,3)//5 for n in range(51)] # G. C. Greubel, Oct 16 2024

Formula

From R. J. Mathar, Apr 15 2010: (Start)
a(n) = +3*a(n-1) -3*a(n-2) +a(n-3) +a(n-5) -3*a(n-6) +3*a(n-7) -a(n-8).
G.f.: x^3*(1+x+3*x^2-x^3+2*x^4) / ( (1-x)^4*(1+x+x^2+x^3+x^4) ). (End)

A011889 a(n) = floor(n*(n-1)*(n-2)/7).

Original entry on oeis.org

0, 0, 0, 0, 3, 8, 17, 30, 48, 72, 102, 141, 188, 245, 312, 390, 480, 582, 699, 830, 977, 1140, 1320, 1518, 1734, 1971, 2228, 2507, 2808, 3132, 3480, 3852, 4251, 4676, 5129, 5610, 6120, 6660, 7230, 7833, 8468, 9137, 9840, 10578, 11352, 12162, 13011, 13898, 14825, 15792, 16800
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(n*(n-1)*(n-2)/7): n in [0..50]]; // Vincenzo Librandi, Jul 07 2012
  • Mathematica
    CoefficientList[Series[x^4*(3-x+2*x^2+x^4+x^5)/((-1+x)^4*(1+x+x^2+x^3+ x^4+x^5+x^6)),{x,0,50}],x] (* Vincenzo Librandi Jul 07 2012 *)
Floor[6*Binomial[Range[0,50], 3]/7] (* G. C. Greubel, Oct 06 2024 *)
  • SageMath
    [6*binomial(n,3)//7 for n in range(51)] # G. C. Greubel, Oct 06 2024

Formula

From R. J. Mathar, Apr 15 2010: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-7) - 3*a(n-8) + 3*a(n-9) - a(n-10).
G.f.: x^4*(3-x+2*x^2+x^4+x^5) / ( (1-x)^4*(1+x+x^2+x^3+x^4+x^5+x^6) ). (End)

A011890 a(n) = floor( n*(n-1)*(n-2)/8 ).

Original entry on oeis.org

0, 0, 0, 0, 3, 7, 15, 26, 42, 63, 90, 123, 165, 214, 273, 341, 420, 510, 612, 726, 855, 997, 1155, 1328, 1518, 1725, 1950, 2193, 2457, 2740, 3045, 3371, 3720, 4092, 4488, 4908, 5355, 5827, 6327, 6854, 7410, 7995, 8610, 9255, 9933, 10642, 11385, 12161, 12972, 13818, 14700
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(6*Binomial(n,3)/8): n in [0..50]]; // G. C. Greubel, Oct 06 2024
  • Mathematica
    Floor[6*Binomial[Range[0,50], 3]/8] (* G. C. Greubel, Oct 06 2024 *)
  • SageMath
    [6*binomial(n,3)//8 for n in range(51)] # G. C. Greubel, Oct 06 2024

Formula

From R. J. Mathar, Apr 15 2010: (Start)
a(n) = +3*a(n-1) -3*a(n-2) +a(n-3) +a(n-8) -3*a(n-9) +3*a(n-10) -a(n-11).
G.f.: x^4*(3-2*x+3*x^2-x^3+2*x^4+x^6)/((1-x)^4*(1+x)*(1+x^2)*(1+x^4)). (End)

Extensions

a(41) onwards from G. C. Greubel, Oct 06 2024

A011891 a(n) = floor(n*(n-1)*(n-2)/9).

Original entry on oeis.org

0, 0, 0, 0, 2, 6, 13, 23, 37, 56, 80, 110, 146, 190, 242, 303, 373, 453, 544, 646, 760, 886, 1026, 1180, 1349, 1533, 1733, 1950, 2184, 2436, 2706, 2996, 3306, 3637, 3989, 4363, 4760, 5180, 5624, 6092, 6586, 7106, 7653, 8227, 8829, 9460, 10120, 10810, 11530
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(n*(n-1)*(n-2)/9): n in [0..50]]; // Vincenzo Librandi, Feb 23 2017
  • Mathematica
    Table[Floor[(n(n-1)(n-2))/9],{n,0,40}] (* or *)
LinearRecurrence[{4,-6,3,3,-6,3,3,-6,4,-1}, {0,0,0,0,2,6,13,23,37,56}, 50] (* Harvey P. Dale, Feb 20 2017 *)
  • SageMath
    [2*binomial(n,3)//3 for n in range(51)] # G. C. Greubel, Oct 06 2024

Formula

G.f.: x^4*(2-2*x+x^2+x^3-x^4+x^5)/((1+x^3+x^6)*(1-x)^4). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009]
Third differences are [-2, 4] repeated. - M. F. Hasler, Sep 15 2009

A011892 a(n) = floor( n*(n-1)*(n-2)/10 ).

Original entry on oeis.org

0, 0, 0, 0, 2, 6, 12, 21, 33, 50, 72, 99, 132, 171, 218, 273, 336, 408, 489, 581, 684, 798, 924, 1062, 1214, 1380, 1560, 1755, 1965, 2192, 2436, 2697, 2976, 3273, 3590, 3927, 4284, 4662, 5061, 5483, 5928, 6396, 6888, 7404, 7946, 8514, 9108, 9729, 10377, 11054, 11760
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(n*(n-1)*(n-2)/10 ): n in [0..50]]; // Vincenzo Librandi, Jul 07 2012
  • Mathematica
    CoefficientList[Series[x^4*(2+x^3)/((1-x)^3*(1-x^5)),{x,0,50}],x] (* Vincenzo Librandi, Jul 07 2012 *)
Floor[3*Binomial[Range[0,50], 3]/5] (* G. C. Greubel, Oct 06 2024 *)
  • SageMath
    Floor[3*Binomial[Range[0, 50], 3]/5] # G. C. Greubel, Oct 06 2024

Formula

From R. J. Mathar, Apr 15 2010: (Start)
a(n) = +3*a(n-1) -3*a(n-2) +a(n-3) +a(n-5) -3*a(n-6) +3*a(n-7) -a(n-8).
G.f.: x^4*(2+x^3)/( (1-x)^4*(1+x+x^2+x^3+x^4) ). (End)

A011893 a(n) = floor( n*(n-1)*(n-2)/11 ).

Original entry on oeis.org

0, 0, 0, 0, 2, 5, 10, 19, 30, 45, 65, 90, 120, 156, 198, 248, 305, 370, 445, 528, 621, 725, 840, 966, 1104, 1254, 1418, 1595, 1786, 1993, 2214, 2451, 2705, 2976, 3264, 3570, 3894, 4238, 4601, 4984, 5389, 5814, 6261, 6731, 7224, 7740, 8280, 8844, 9434, 10049, 10690
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(6*Binomial(n,3)/11): n in [0..50]]; // G. C. Greubel, Oct 06 2024
  • Mathematica
    Table[Floor[n(n-1)(n-2)/11],{n,0,40}] (* or *)
LinearRecurrence[{3,-3,1,0,0,0,0,0,0,0,1,-3,3,-1}, {0,0,0,0,2,5,10,19,30,45, 65,90,120,156}, 50] (* Harvey P. Dale, Nov 23 2018 *)
  • SageMath
    [6*binomial(n,3)//11 for n in range(51)] # G. C. Greubel, Oct 06 2024

Formula

a(n) = +3*a(n-1) -3*a(n-2) +a(n-3) +a(n-11) -3*a(n-12) +3*a(n-13) -a(n-14). - R. J. Mathar, Apr 15 2010
G.f.: x^4*(2-x+x^2+2*x^3-2*x^4+2*x^5+x^6+x^9)/((1-x)^4*(1+x+x^2+x^3+x^4+x^5 +x^6+x^7+x^8+x^9+x^10)). - Peter J. C. Moses, Jun 02 2014

Extensions

a(41) onwards from G. C. Greubel, Oct 06 2024

A011894 a(n) = floor(n*(n-1)*(n-2)/12).

Original entry on oeis.org

0, 0, 0, 0, 2, 5, 10, 17, 28, 42, 60, 82, 110, 143, 182, 227, 280, 340, 408, 484, 570, 665, 770, 885, 1012, 1150, 1300, 1462, 1638, 1827, 2030, 2247, 2480, 2728, 2992, 3272, 3570, 3885, 4218, 4569, 4940, 5330, 5740, 6170, 6622, 7095, 7590, 8107, 8648, 9212, 9800
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n+1) = floor((n^3-n)/12) is an upper bound for the Kirchhoff index of a circulant graph with n vertices [Zhang & Yang]. - R. J. Mathar, Apr 26 2007
Also the matching number of the n-tetrahedral graph. - Eric W. Weisstein, Jun 20 2017

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(n*(n-1)*(n-2)/12): n in [0..50]]; // Vincenzo Librandi, Jul 07 2012
  • Maple
    seq(floor(binomial(n,3)/2), n=0..40); # Zerinvary Lajos, Jan 12 2009
  • Mathematica
    CoefficientList[Series[x^4*(2-x+x^2)/((1-x)^3*(1-x^4)),{x, 0, 50}], x] (* Vincenzo Librandi, Jul 07 2012 *)
(* Contributions from Eric W. Weisstein, Jun 20 2017 *)
Table[(3*((-1)^n -1) + 2*n*(n-1)*(n-2) + 6*Sin[(n*Pi)/2])/24, {n,0,50}]
LinearRecurrence[{3,-3,1,1,-3,3,-1}, {0,0,0,2,5,10,17}, 50] (* End *)
Floor[Binomial[Range[0,50], 3]/2] (* G. C. Greubel, Oct 06 2024 *)
  • Sage
    [floor(binomial(n,3)/2) for n in range(41)] # Zerinvary Lajos, Dec 01 2009

Formula

From R. J. Mathar, Apr 15 2010: (Start)
a(n) = +3*a(n-1) -3*a(n-2) +a(n-3) +a(n-4) -3*a(n-5) +3*a(n-6) -a(n-7).
G.f.: x^4*(2-x+x^2) / ( (1-x)^4*(1+x)*(1+x^2) ). (End)
a(n) = (1/24)*(2*n^3 - 6*n^2 + 4*n - 3*(1-(-1)^n)*(1 - (-1)^((2*n-1+(-1)^n)/4)) ). - Luce ETIENNE, Jun 26 2014

A011897 a(n) = floor(n*(n-1)*(n-2)/15).

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 8, 14, 22, 33, 48, 66, 88, 114, 145, 182, 224, 272, 326, 387, 456, 532, 616, 708, 809, 920, 1040, 1170, 1310, 1461, 1624, 1798, 1984, 2182, 2393, 2618, 2856, 3108, 3374, 3655, 3952, 4264, 4592, 4936, 5297, 5676, 6072, 6486, 6918, 7369, 7840
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(n*(n-1)*(n-2)/15): n in [0..50]]; // Vincenzo Librandi, Jul 07 2012
  • Mathematica
    CoefficientList[Series[x^4(1+x-x^2+x^3)/((1-x)^3*(1-x^5)),{x,0,45}],x]  (* Harvey P. Dale, Feb 25 2011 *)
Floor[2*Binomial[Range[0, 50], 3]/5] (* G. C. Greubel, Oct 16 2024 *)
  • SageMath
    [2*binomial(n,3)//5 for n in range(51)] # G. C. Greubel, Oct 16 2024

Formula

From R. J. Mathar, Apr 15 2010: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5) - 3*a(n-6) + 3*a(n-7) - a(n-8).
G.f.: x^4*(1+x-x^2+x^3) / ( (1-x)^4*(1+x+x^2+x^3+x^4) ). (End)
Showing 1-10 of 25 results. Next

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

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