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.

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A011898 a(n) = floor(n*(n-1)*(n-2)/16).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 7, 13, 21, 31, 45, 61, 82, 107, 136, 170, 210, 255, 306, 363, 427, 498, 577, 664, 759, 862, 975, 1096, 1228, 1370, 1522, 1685, 1860, 2046, 2244, 2454, 2677, 2913, 3163, 3427, 3705, 3997, 4305
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(n*(n-1)*(n-2)/16): n in [0..50]]; // Vincenzo Librandi, May 21 2012
  • Mathematica
    Table[Floor[(n(n-1)(n-2))/16],{n,0,50}] (* Harvey P. Dale, May 16 2012 *)
  • PARI
    a(n) = n*(n-1)*(n-2)\16; \\ Michel Marcus, Jan 14 2018
  • SageMath
    [3*binomial(n,3)//8 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-16) - 3*a(n-17) + 3*a(n-18) - a(n-19).
G.f.: x^4*(1+x^2+2*x^6-2*x^7+3*x^8-x^9+x^11+x^12-x^13+x^14) / ( (1-x)^4*(1+x)*(1+x^2)*(1+x^4)*(1+x^8) ). (End)

A011899 a(n) = floor(n*(n-1)*(n-2)/17).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 7, 12, 19, 29, 42, 58, 77, 100, 128, 160, 197, 240, 288, 342, 402, 469, 543, 625, 714, 811, 917, 1032, 1156, 1289, 1432, 1586, 1750, 1925, 2112, 2310, 2520, 2742, 2977, 3225, 3487, 3762, 4051
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A007531, A011886.

Programs

Formula

G.f.: (x^4*(1+x^2-x^3+x^4+x^5+x^9+x^10-x^11+x^12+x^13-x^14+x^15))/(1+x*(-3+x*(3+x*(-1+(-1+x)^3*x^14)))). - Peter J. C. Moses, Jun 02 2014

A011901 a(n) = floor( n*(n-1)*(n-2)/19 ).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 6, 11, 17, 26, 37, 52, 69, 90, 114, 143, 176, 214, 257, 306, 360, 420, 486, 559, 639, 726, 821, 923, 1034, 1153, 1282, 1419, 1566, 1722, 1889, 2066, 2254, 2453, 2664, 2886, 3120, 3366, 3625, 3897, 4182, 4481, 4793, 5120, 5461, 5818, 6189, 6576, 6978, 7397, 7832, 8284, 8753, 9240, 9744, 10266, 10806, 11365, 11943, 12540, 13157, 13793, 14450, 15127
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(6*Binomial(n,3)/19): n in [0..75]]; // G. C. Greubel, Oct 17 2024
  • Mathematica
    Table[Floor[n(n-1)(n-2)/19],{n,0,75}] (* or  *)
LinearRecurrence[{3,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1}, {0,0,0,0, 1,3,6,11,17,26,37,52,69,90,114,143,176,214,257,306,360,420}, 76] (* Harvey P. Dale, May 30 2021 *)
  • PARI
    a(n)=n*(n-1)*(n-2)\19 \\ Charles R Greathouse IV, Oct 21 2022
  • Python
    print([6*binomial(n,3)//19 for n in range(76)]) # G. C. Greubel, Oct 17 2024

Formula

a(n) = +3*a(n-1) -3*a(n-2) +a(n-3) +a(n-19) -3*a(n-20) +3*a(n-21) -a(n-22). - R. J. Mathar, Apr 15 2010
G.f.: x^4 (1+x^4) (1+x^3 (1-x+x^3+x^6) (1+(-1+x) x (1+x^2)))/(1+x (-3+x (3+x (-1+(-1+x)^3 x^16)))). - Peter J. C. Moses, Jun 02 2014
G.f.: x^4*(1+x^4)*(1+x^3-2*x^4+2*x^5-x^6+x^7+x^11-x^12+x^13)/((1-x)^3*(1-x^19)). - G. C. Greubel, Oct 17 2024

Extensions

More terms added by G. C. Greubel, Oct 18 2024

A011902 a(n) = floor( n*(n-1)*(n-2)/20 ).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 6, 10, 16, 25, 36, 49, 66, 85, 109, 136, 168, 204, 244, 290, 342, 399, 462, 531, 607, 690, 780, 877, 982, 1096, 1218, 1348, 1488, 1636, 1795, 1963, 2142, 2331, 2530, 2741, 2964, 3198, 3444, 3702, 3973, 4257, 4554, 4864, 5188, 5527, 5880, 6247, 6630, 7027, 7441, 7870, 8316, 8778, 9256, 9752, 10266, 10797, 11346, 11913, 12499, 13104, 13728, 14371
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(3*Binomial(n,3)/10): n in [0..80]]; // G. C. Greubel, Oct 18 2024
  • Mathematica
    Table[Floor[(n(n-1)(n-2))/20],{n,0,80}]  (* Harvey P. Dale, Mar 23 2011 *)
  • SageMath
    [3*binomial(n,3)//10 for n in range(81)] # G. C. Greubel, Oct 18 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-20) -3*a(n-21) +3*a(n-22) -a(n-23).
G.f.: x^4*(1+x^4+x^5-x^6+2*x^8-2*x^9+3*x^10-2*x^11+2*x^12-x^13+2*x^15-x^17+x^18)/((1-x)^4*(1+x)*(1+x^2)*(1+x+x^2+x^3+x^4)*(1-x+x^2-x^3+x^4)*(1-x^2+x^4-x^6+x^8)). (End)

Extensions

More terms added by G. C. Greubel, Oct 18 2024

A011903 a(n) = floor(n*(n-1)*(n-2)/21).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 10, 16, 24, 34, 47, 62, 81, 104, 130, 160, 194, 233, 276, 325, 380, 440, 506, 578, 657, 742, 835, 936, 1044, 1160, 1284, 1417, 1558, 1709, 1870, 2040, 2220, 2410, 2611, 2822, 3045, 3280
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(n*(n-1)*(n-2)/21): n in [0..50]]; // Vincenzo Librandi, Jul 07 2012
  • Mathematica
    CoefficientList[Series[x^4*(1-x+2*x^2-x^4+x^5)/((1-x)^3*(1-x^7)), {x,0,60}], x] (* Vincenzo Librandi, Jul 07 2012 *)
LinearRecurrence[{3,-3,1,0,0,0,1,-3,3,-1},{0,0,0,0,1,2,5,10,16,24},60] (* Harvey P. Dale, May 03 2023 *)
  • SageMath
    [2*binomial(n,3)//7 for n in range(61)] # G. C. Greubel, Oct 18 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*(1-x+2*x^2-x^4+x^5) / ( (1-x)^4*(1+x+x^2+x^3+x^4+x^5+x^6) ). (End)

A011904 a(n) = floor( n*(n-1)*(n-2)/22 ).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 9, 15, 22, 32, 45, 60, 78, 99, 124, 152, 185, 222, 264, 310, 362, 420, 483, 552, 627, 709, 797, 893, 996, 1107, 1225, 1352, 1488, 1632, 1785, 1947, 2119, 2300, 2492, 2694, 2907, 3130, 3365, 3612, 3870, 4140, 4422, 4717, 5024, 5345, 5679, 6027, 6388, 6764, 7155, 7560, 7980, 8415, 8866, 9332, 9815, 10314, 10830, 11362, 11912, 12480, 13065, 13668
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(3*Binomial(n,3)/11): n in [0..80]]; // G. C. Greubel, Oct 18 2024
  • Mathematica
    Table[Floor[(n(n-1)(n-2))/22],{n,0,80}] (* Harvey P. Dale, Sep 30 2019 *)
  • SageMath
    [3*binomial(n,3)//11 for n in range(81)] # G. C. Greubel, Oct 18 2024

Formula

G.f.: x^4*(1-x+2*x^2-x^3+x^4-x^5+2*x^6-x^8+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

More terms added by G. C. Greubel, Oct 18 2024

A011905 a(n) = floor( n*(n-1)*(n-2)/23 ).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 9, 14, 21, 31, 43, 57, 74, 94, 118, 146, 177, 212, 252, 297, 346, 401, 462, 528, 600, 678, 763, 854, 953, 1059, 1172, 1293, 1423, 1561, 1707, 1862, 2026, 2200, 2384, 2577, 2780, 2994, 3219, 3454, 3701, 3960, 4230, 4512, 4806, 5113, 5432, 5765, 6111, 6470, 6843, 7231, 7633, 8049, 8480, 8926, 9388, 9866, 10359, 10868, 11394, 11937, 12496, 13073
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(6*Binomial(n,3)/23): n in [0..75]]; // G. C. Greubel, Oct 18 2024
  • Mathematica
    Table[Floor[(n(n-1)(n-2))/23],{n,0,75}] (* Harvey P. Dale, Aug 06 2013 *)
  • SageMath
    [6*binomial(n,3)//23 for n in range(76)] # G. C. Greubel, Oct 18 2024

Formula

G.f.: x^4*(1+x*(-1+x*(2+x*(-1+x^2*(1+(-1+x)*x^2)*(1+x+x^6+x^10+x^13))))) /(1+x*(-3+x*(3+x*(-1+(-1+x)^3*x^20)))). - Peter J. C. Moses, Jun 02 2014

Extensions

More terms added by G. C. Greubel, Oct 18 2024

A011907 a(n) = floor( n*(n-1)*(n-2)/25 ).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 4, 8, 13, 20, 28, 39, 52, 68, 87, 109, 134, 163, 195, 232, 273, 319, 369, 425, 485, 552, 624, 702, 786, 876, 974, 1078, 1190, 1309, 1436, 1570, 1713, 1864, 2024, 2193, 2371, 2558, 2755, 2961, 3178, 3405, 3643, 3891, 4151, 4421, 4704, 4998, 5304, 5622, 5952, 6296, 6652, 7022, 7405, 7802, 8212, 8637, 9076, 9530, 9999, 10483, 10982, 11497, 12027
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

Formula

From R. J. Mathar, Apr 15 2010: (Start)
a(n) = +3*a(n-1) -3*a(n-2) +a(n-3) +a(n-25) -3*a(n-26) +3*a(n-27) -a(n-28).
G.f.: x^5*(2-2*x+2*x^2-x^3+x^4-x^5+2*x^6-x^7+x^8+x^12-x^13+2*x^14-x^15+x^16-x^17+2*x^18-2*x^19+3*x^20-2*x^21+x^22) / ( (1-x)^4*(1+x^4+x^3+x^2+x)*(1+x^5+x^10+x^15+x^20) ). (End)

Extensions

More terms added by G. C. Greubel, Oct 18 2024

A011908 a(n) = floor( n*(n-1)*(n-2)/26 ).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 4, 8, 12, 19, 27, 38, 50, 66, 84, 105, 129, 156, 188, 223, 263, 306, 355, 408, 467, 530, 600, 675, 756, 843, 936, 1037, 1144, 1259, 1380, 1510, 1647, 1793, 1946, 2109, 2280, 2460, 2649, 2847, 3056, 3274, 3503, 3741, 3991, 4251, 4523, 4805, 5100, 5406, 5724, 6054, 6396, 6752, 7120
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(3*Binomial(n,3)/13): n in [0..75]]; // G. C. Greubel, Oct 18 2024
  • Mathematica
    Floor[3*Binomial[Range[0, 75], 3]/13] (* G. C. Greubel, Oct 18 2024 *)
  • SageMath
    [3*binomial(n,3)//13 for n in range(76)] # G. C. Greubel, Oct 18 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-13) -3*a(n-14) +3*a(n-15) -a(n-16).
G.f.: x^5*(2-2*x+2*x^2-2*x^3+3*x^4-2*x^5+2*x^6-2*x^7+3*x^8-2*x^9+x^10) / ( (1-x)^3*(1-x^13) ). (End)

Extensions

More terms added by G. C. Greubel, Oct 18 2024

A011909 a(n) = floor( n*(n-1)*(n-2)/27 ).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 4, 7, 12, 18, 26, 36, 48, 63, 80, 101, 124, 151, 181, 215, 253, 295, 342, 393, 449, 511, 577, 650, 728, 812, 902, 998, 1102, 1212, 1329, 1454, 1586, 1726, 1874, 2030, 2195, 2368, 2551, 2742, 2943, 3153, 3373, 3603, 3843, 4094, 4355, 4627, 4911, 5205, 5512, 5830, 6160, 6502, 6856, 7224, 7604, 7997, 8404, 8824, 9258, 9706, 10168, 10645, 11136
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011886.

Programs

  • Magma
    [Floor(2*Binomial(n,3)/9): n in [0..80]]; // G. C. Greubel, Oct 19 2024
  • Mathematica
    Table[Floor[n(n-1)(n-2)/27],{n,0,80}] (* or *)
LinearRecurrence[{4,-6,3,3,-6,3,3,-6,3,3,-6,3,3,-6,3,3,-6,3,3,-6,3,3,-6,3,3,-6,4,-1},{0,0,0,0,0,2,4,7,12,18,26,36,48,63,80,101,124,151,181,215,253,295, 342,393,449,511,577,650}, 81] (* Harvey P. Dale, Jun 12 2023 *)
  • SageMath
    [2*binomial(n,3)//9 for n in range(81)] # G. C. Greubel, Oct 19 2024

Formula

G.f.: x^5*(1-x+x^2)*(2-2*x-x^2+3*x^3-2*x^4+3*x^6-3*x^7+2*x^9-x^10-x^11 +3*x^12-2*x^13-x^14+3*x^15-2*x^16+2*x^18-2*x^19+x^20)/((1-x)^4*(1+x^3+x^6)*(1+x^9+x^18)). - Peter J. C. Moses, Jun 02 2014

Extensions

More terms added by G. C. Greubel, Oct 19 2024
Previous Showing 11-20 of 25 results. Next

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

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