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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A212099 Number of (w,x,y,z) with all terms in {1,...,n} and w^3>x^3+y^3+z^3.

Original entry on oeis.org

0, 0, 1, 9, 35, 89, 192, 367, 645, 1047, 1620, 2395, 3414, 4749, 6435, 8518, 11079, 14171, 17876, 22272, 27409, 33396, 40290, 48249, 57265, 67548, 79146, 92127, 106708, 122880, 140876, 160757, 182694, 206791, 233160, 262032, 293445
Offset: 0

Views

Author

Clark Kimberling, May 03 2012

Keywords

Comments

a(n)+A212098(n)=n^4. For a guide to related sequences, see A211795.

Links

Crossrefs

Cf. A211795, A212098.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^3 > x^3 + y^3 + z^3, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A212099 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212100 Number of (w,x,y,z) with all terms in {1,...,n} and w^3>=x^3+y^3+z^3.

Original entry on oeis.org

0, 0, 1, 9, 35, 89, 198, 373, 651, 1059, 1632, 2407, 3432, 4767, 6453, 8536, 11097, 14189, 17906, 22308, 27451, 33438, 40332, 48291, 57313, 67602, 79200, 92187, 106774, 122952, 140954, 160835, 182772, 206869, 233238, 262110, 293535
Offset: 0

Views

Author

Clark Kimberling, May 03 2012

Keywords

Comments

a(n)+A212097(n)=n^4. For a guide to related sequences, see A211795.

Crossrefs

Cf. A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^3 >= x^3 + y^3 + z^3, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A212100 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212104 Number of (w,x,y,z) with all terms in {1,...,n} and w <= harmonic mean of {x,y,z}.

Original entry on oeis.org

0, 1, 9, 36, 106, 252, 528, 964, 1617, 2559, 3880, 5631, 7950, 10900, 14595, 19161, 24727, 31419, 39399, 48790, 59799, 72570, 87277, 104124, 123342, 145075, 169575, 197061, 227779, 261915, 299778, 341599, 387624, 438171, 493486
Offset: 0

Views

Author

Clark Kimberling, May 04 2012

Keywords

Comments

a(n)+A212105(n)=n^4.
A 4-tuple (w,x,y,z) is counted if 3/w<=1/x+1/y+1/z.
For a guide to related sequences, see A211795.

Crossrefs

Cf. A211795, A212103.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*(y*z + z*x + x*y) <= 3 x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212104 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)

A212105 Number of (w,x,y,z) with all terms in {1,...,n} and w > harmonic mean of {x,y,z}.

Original entry on oeis.org

0, 0, 7, 45, 150, 373, 768, 1437, 2479, 4002, 6120, 9010, 12786, 17661, 23821, 31464, 40809, 52102, 65577, 81531, 100201, 121911, 146979, 175717, 208434, 245550, 287401, 334380, 386877, 445366, 510222, 581922, 660952, 747750, 842850
Offset: 0

Views

Author

Clark Kimberling, May 04 2012

Keywords

Comments

a(n)+A212104(n)=n^4.
A 4-tuple (w,x,y,z) is counted if 3/w>1/x+1/y+1/z.
For a guide to related sequences, see A211795.

Crossrefs

Cf. A211795, A212103.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*(y*z + z*x + x*y) > 3 x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A212105 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212106 Number of (w,x,y,z) with all terms in {1,...,n} and w < harmonic mean of {x,y,z}.

Original entry on oeis.org

0, 0, 7, 33, 96, 241, 498, 933, 1579, 2520, 3828, 5578, 7866, 10815, 14509, 19044, 24603, 31294, 39255, 48645, 59599, 72345, 87051, 103897, 123060, 144792, 169291, 196776, 227445, 261580, 299358, 341178, 387196, 437736, 493050
Offset: 0

Views

Author

Clark Kimberling, May 04 2012

Keywords

Comments

a(n)+A212107(n)=n^4.
A 4-tuple (w,x,y,z) is counted if 3/w<1/x+1/y+1/z.
For a guide to related sequences, see A211795.

Crossrefs

Cf. A211795, A212103.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*(y*z + z*x + x*y) < 3 x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A212106 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212107 Number of (w,x,y,z) with all terms in {1,...,n} and w >= harmonic mean of {x,y,z}.

Original entry on oeis.org

0, 1, 9, 48, 160, 384, 798, 1468, 2517, 4041, 6172, 9063, 12870, 17746, 23907, 31581, 40933, 52227, 65721, 81676, 100401, 122136, 147205, 175944, 208716, 245833, 287685, 334665, 387211, 445701, 510642, 582343, 661380, 748185, 843286
Offset: 0

Views

Author

Clark Kimberling, May 04 2012

Keywords

Comments

a(n)+A212106(n)=n^4.
A 4-tuple (w,x,y,z) is counted if 3/w>=1/x+1/y+1/z.
For a guide to related sequences, see A211795.

Crossrefs

Cf. A211795, A212103.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*(y*z + z*x + x*y) >= 3 x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A212107 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212134 Number of (w,x,y,z) with all terms in {1,...,n} and median<=mean.

Original entry on oeis.org

0, 1, 12, 57, 172, 405, 816, 1477, 2472, 3897, 5860, 8481, 11892, 16237, 21672, 28365, 36496, 46257, 57852, 71497, 87420, 105861, 127072, 151317, 178872, 210025, 245076, 284337, 328132, 376797, 430680, 490141, 555552, 627297, 705772, 791385, 884556, 985717
Offset: 0

Views

Author

Clark Kimberling, May 04 2012

Keywords

Comments

Also, the number of (w,x,y,z) with all terms in {1,...,n} and median>=mean.
For a guide to related sequences, see A211795.

Links

Crossrefs

Cf. A211795, A212133.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[Apply[Plus, Rest[Most[Sort[{w, x, y, z}]]]]/2 <= (w + x + y + z)/4, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #},
{z, 1, #}] &[n]; s)]];
Flatten[Map[{t[#]} &, Range[0, 50]]]  (* A212134 *)
(* Peter J. C. Moses, May 01 2012 *)
  • PARI
    concat(0, Vec(x*(1 + 7*x + 7*x^2 - 3*x^3) /(1 - x)^5 + O(x^40))) \\ Colin Barker, Dec 02 2017

Formula

a(n)+ A212135(n) = n^4.
a(n) = n*(n^3 + 2*n^2 - 3*n + 2)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1 + 7*x + 7*x^2 - 3*x^3) /(1 - x)^5. - Colin Barker, Dec 02 2017
E.g.f.: exp(x)*x*(2 + 10*x + 8*x^2 + x^3)/2. - Stefano Spezia, Aug 08 2025

A212247 Number of (w,x,y,z) with all terms in {1,...,n} and 3w=x+y+z+n.

Original entry on oeis.org

0, 1, 4, 13, 29, 56, 95, 150, 222, 315, 430, 571, 739, 938, 1169, 1436, 1740, 2085, 2472, 2905, 3385, 3916, 4499, 5138, 5834, 6591, 7410, 8295, 9247, 10270, 11365, 12536, 13784, 15113, 16524, 18021, 19605, 21280, 23047, 24910, 26870, 28931
Offset: 0

Views

Author

Clark Kimberling, May 09 2012

Keywords

Comments

For a guide to related sequences, see A211795.

Links

Crossrefs

Cf. A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 w == x + y + z - n, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 60]]  (* A212246 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{3, -2, -2, 3, -1},{0, 1, 4, 13, 29},42] (* Ray Chandler, Aug 02 2015 *)
CoefficientList[Series[x (1+x+3x^2)/((1+x)(1-x)^4),{x,0,50}],x] (* Harvey P. Dale, Jul 06 2021 *)

Formula

a(n) = 3*a(n-1)-3*a(n-2)+2*a(n-3)-3*a(n-4)+3*a(n-5)-a(n-6).
G.f.: x*(1+x+3*x^2)/((1+x)*(1-x)^4). [Bruno Berselli, May 30 2012]
a(n) = (2*n*(10*n^2+3*n+2)-9(-1)^n+9)/48. [Bruno Berselli, May 30 2012]

A212249 Number of (w,x,y,z) with all terms in {1,...,n} and 3w

Original entry on oeis.org

0, 1, 12, 63, 202, 496, 1034, 1923, 3289, 5280, 8062, 11820, 16761, 23110, 31111, 41030, 53151, 67777, 85233, 105862, 130026, 158109, 190513, 227659, 269990, 317967, 372070, 432801, 500680, 576246, 660060, 752701, 854767, 966878
Offset: 0

Views

Author

Clark Kimberling, May 09 2012

Keywords

Comments

a(n)+A212250(n) = n^4.
For a guide to related sequences, see A211795.

Links

Crossrefs

Cf. A211795, A212247.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[3 w < x + y + z + n, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]  (* A212249  *)
(* Peter J. C. Moses, Apr 13 2012 *)

Formula

a(n) = 4*a(n-1)-6*a(n-2)+5*a(n-3)-5*a(n-4)+6*a(n-5)-4*a(n-6)+a(n-7).
G.f.: x*(1+8*x+21*x^2+17*x^3+11*x^4+x^5)/((1+x+x^2)*(1-x)^5). - Bruno Berselli, Jun 05 2012
a(n) = (59*n^4 -10*n^3 +5*n^2 -6*n -8*((((n+1) mod 3) +(-1)^((n+1) mod 3))*(-1)^(n mod 3)))/72. - Bruno Berselli, Jun 05 2012

A212250 Number of (w,x,y,z) with all terms in {1,...,n} and 3w>=x+y+z+n.

Original entry on oeis.org

0, 0, 4, 18, 54, 129, 262, 478, 807, 1281, 1938, 2821, 3975, 5451, 7305, 9595, 12385, 15744, 19743, 24459, 29974, 36372, 43743, 52182, 61786, 72658, 84906, 98640, 113976, 131035, 149940, 170820, 193809, 219043, 246664, 276819, 309657
Offset: 0

Views

Author

Clark Kimberling, May 09 2012

Keywords

Comments

a(n)+A212249 = n^4.
For a guide to related sequences, see A211795.

Links

Crossrefs

Cf. A211795, A212247, A212249.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[3 w >= x + y + z + n, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]  (* A212250 *)
(* Peter J. C. Moses, Apr 13 2012 *)

Formula

a(n) = 4*a(n-1)-6*a(n-2)+5*a(n-3)-5*a(n-4)+6*a(n-5)-4*a(n-6)+a(n-7).
G.f.: x^2*(4+2*x+6*x^2+x^3)/((1+x+x^2)*(1-x)^5). - Bruno Berselli, Jun 05 2012
a(n) = (13*n^4+10*n^3-5*n^2+6*n+8*b)/72, where b = 0,-3,1,0,-3,1,... (repeated). [Bruno Berselli, Jun 05 2012]
Previous Showing 51-60 of 202 results. Next

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