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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A211546 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w=3x-3y.

Original entry on oeis.org

0, 0, 0, 2, 3, 4, 9, 11, 13, 21, 24, 27, 38, 42, 46, 60, 65, 70, 87, 93, 99, 119, 126, 133, 156, 164, 172, 198, 207, 216, 245, 255, 265, 297, 308, 319, 354, 366, 378, 416, 429, 442, 483, 497, 511, 555, 570, 585, 632, 648, 664, 714, 731, 748, 801, 819
Offset: 0

Views

Author

Clark Kimberling, Apr 15 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[w - 3 x + 3 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211546 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,0,2,-2,0,-1,1},{0,0,0,2,3,4,9},56] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(vector(3), Vec(x^3*(2 + x + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^40))) \\ Colin Barker, Dec 03 2017

Formula

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7).
G.f.: x^3*(2 + x + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2)^2). - Colin Barker, Dec 03 2017

A211612 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+x+y>=0.

Original entry on oeis.org

0, 4, 35, 117, 274, 530, 909, 1435, 2132, 3024, 4135, 5489, 7110, 9022, 11249, 13815, 16744, 20060, 23787, 27949, 32570, 37674, 43285, 49427, 56124, 63400, 71279, 79785, 88942, 98774, 109305, 120559, 132560, 145332, 158899, 173285, 188514, 204610, 221597
Offset: 0

Views

Author

Clark Kimberling, Apr 16 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t = Compile[{{u, _Integer}}, Module[{s = 0}, (Do[If[w + x + y >= 0, s = s + 1], {w, #}, {x, #}, {y, #}] &[Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211612 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{4, -6, 4, -1},{0, 4, 35, 117},36] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(4 + 19*x + x^2) / (1 - x)^4 + O(x^40))) \\ Colin Barker, Dec 04 2017

Formula

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
From Colin Barker, Dec 04 2017: (Start)
G.f.: x*(4 + 19*x + x^2) / (1 - x)^4.
a(n) = (n*(-3 + 3*n + 8*n^2))/2.
(End)

A211613 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+x+y>1.

Original entry on oeis.org

0, 1, 20, 78, 199, 407, 726, 1180, 1793, 2589, 3592, 4826, 6315, 8083, 10154, 12552, 15301, 18425, 21948, 25894, 30287, 35151, 40510, 46388, 52809, 59797, 67376, 75570, 84403, 93899, 104082, 114976, 126605, 138993, 152164, 166142, 180951, 196615, 213158
Offset: 0

Views

Author

Clark Kimberling, Apr 16 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t = Compile[{{u, _Integer}}, Module[{s = 0}, (Do[If[w + x + y > 1, s = s + 1], {w, #}, {x, #}, {y, #}] &[Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211613 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
Join[{0},LinearRecurrence[{4, -6, 4, -1},{1, 20, 78, 199},35]] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(1 + 16*x + 4*x^2 + 3*x^3) / (1 - x)^4 + O(x^40))) \\ Colin Barker, Dec 04 2017

Formula

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
From Colin Barker, Dec 04 2017: (Start)
G.f.: x*(1 + 16*x + 4*x^2 + 3*x^3) / (1 - x)^4.
a(n) = (-6 + 9*n - 9*n^2 + 8*n^3)/2 for n > 0. (End)
E.g.f.: 3 + exp(x)*(4*x^3 + 15*x^2/2 + 4*x - 3). - Stefano Spezia, Jun 20 2025

A211614 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+x+y>2.

Original entry on oeis.org

0, 1, 11, 57, 160, 344, 633, 1051, 1622, 2370, 3319, 4493, 5916, 7612, 9605, 11919, 14578, 17606, 21027, 24865, 29144, 33888, 39121, 44867, 51150, 57994, 65423, 73461, 82132, 91460, 101469, 112183, 123626, 135822, 148795, 162569, 177168, 192616, 208937
Offset: 0

Views

Author

Clark Kimberling, Apr 16 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t = Compile[{{u, _Integer}},
   Module[{s = 0}, (Do[If[w + x + y > 2, s = s + 1], {w, #}, {x, #}, {y, #}] &[ Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211614 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
Join[{0, 1},LinearRecurrence[{4, -6, 4, -1},{11, 57, 160, 344},34]] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(1 + 7*x + 19*x^2 - 6*x^3 + 3*x^4) / (1 - x)^4 + O(x^40))) \\ Colin Barker, Dec 04 2017

Formula

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>5.
From Colin Barker, Dec 04 2017: (Start)
G.f.: x*(1 + 7*x + 19*x^2 - 6*x^3 + 3*x^4) / (1 - x)^4.
a(n) = (8*n^3 - 15*n^2 + 15*n - 12)/2 for n>1.
(End)

A211615 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and -1<=w+x+y<=1.

Original entry on oeis.org

0, 6, 24, 60, 114, 186, 276, 384, 510, 654, 816, 996, 1194, 1410, 1644, 1896, 2166, 2454, 2760, 3084, 3426, 3786, 4164, 4560, 4974, 5406, 5856, 6324, 6810, 7314, 7836, 8376, 8934, 9510, 10104, 10716, 11346, 11994, 12660, 13344, 14046
Offset: 0

Views

Author

Clark Kimberling, Apr 16 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t = Compile[{{u, _Integer}}, Module[{s = 0}, (Do[If[-1 <= w + x + y <= 1, s = s + 1], {w, #}, {x, #}, {y, #}] &[Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &,  Range[0, 70]]  (* A211615 *)
%/6  (* A005448 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
Join[{0},LinearRecurrence[{3, -3, 1},{6, 24, 60},40]] (* Ray Chandler, Aug 02 2015 *)

Formula

a(n)= 6*A005448(n).
a(n) = 3a(n-1)-3a(n-2)+a(n-3) for n>3.
a(n) = 6-9*n+9*n^2 for n>0. G.f.: 6*x*(1+x+x^2)/(1-x)^3. [Colin Barker, Sep 09 2012]

A211616 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and -2<=w+x+y<=2.

Original entry on oeis.org

0, 6, 42, 102, 192, 312, 462, 642, 852, 1092, 1362, 1662, 1992, 2352, 2742, 3162, 3612, 4092, 4602, 5142, 5712, 6312, 6942, 7602, 8292, 9012, 9762, 10542, 11352, 12192, 13062, 13962, 14892, 15852, 16842, 17862, 18912, 19992, 21102, 22242, 23412, 24612, 25842
Offset: 0

Views

Author

Clark Kimberling, Apr 16 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t = Compile[{{u, _Integer}}, Module[{s = 0}, (Do[If[-2 <= w + x + y <= 2, s = s + 1], {w, #}, {x, #}, {y, #}] &[Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 70]]  (* A211616 *)
%/6                        (* integers *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
Join[{0, 6},LinearRecurrence[{3, -3, 1},{42, 102, 192},38]] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(6*x*(1 + 4*x - x^2 + x^3) / (1 - x)^3 + O(x^40))) \\ Colin Barker, Dec 04 2017

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.
From Colin Barker, Dec 04 2017: (Start)
G.f.: 6*x*(1 + 4*x - x^2 + x^3) / (1 - x)^3.
a(n) = 3*(4 - 5*n + 5*n^2) for n>1.
(End)

A211617 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and 2w+x+y>0.

Original entry on oeis.org

0, 3, 30, 101, 244, 479, 834, 1329, 1992, 2843, 3910, 5213, 6780, 8631, 10794, 13289, 16144, 19379, 23022, 27093, 31620, 36623, 42130, 48161, 54744, 61899, 69654, 78029, 87052, 96743, 107130, 118233, 130080, 142691, 156094, 170309, 185364, 201279, 218082
Offset: 0

Views

Author

Clark Kimberling, Apr 16 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t = Compile[{{u, _Integer}}, Module[{s = 0}, (Do[If[2 w + x + y > 0, s = s + 1], {w, #}, {x, #}, {y, #}] &[ Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 70]]  (* A211617 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{3, -2, -2, 3, -1},{0, 3, 30, 101, 244},36] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(3 + 21*x + 17*x^2 + 7*x^3) / ((1 - x)^4*(1 + x)) + O(x^40))) \\ Colin Barker, Dec 04 2017

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) for n>4.
From Colin Barker, Dec 04 2017: (Start)
G.f.: x*(3 + 21*x + 17*x^2 + 7*x^3) / ((1 - x)^4*(1 + x)).
a(n) = 4*n^3 - n^2 + n for n even.
a(n) = 4*n^3 - n^2 + n - 1 for n odd.
(End)

A211618 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and 2w+x+y>1.

Original entry on oeis.org

0, 3, 24, 89, 218, 439, 772, 1245, 1878, 2699, 3728, 4993, 6514, 8319, 10428, 12869, 15662, 18835, 22408, 26409, 30858, 35783, 41204, 47149, 53638, 60699, 68352, 76625, 85538, 95119, 105388, 116373, 128094, 140579, 153848, 167929, 182842, 198615, 215268
Offset: 0

Views

Author

Clark Kimberling, Apr 16 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t = Compile[{{u, _Integer}},
   Module[{s = 0}, (Do[If[2 w + x + y > 1,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 70]]  (* A211618 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
Join[{0},LinearRecurrence[{3, -2, -2, 3, -1},{3, 24, 89, 218, 439},35]] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(3 + 15*x + 23*x^2 + 5*x^3 + 2*x^4) / ((1 - x)^4*(1 + x)) + O(x^40))) \\ Colin Barker, Dec 04 2017

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) for n>5.
From Colin Barker, Dec 04 2017: (Start)
G.f.: x*(3 + 15*x + 23*x^2 + 5*x^3 + 2*x^4) / ((1 - x)^4*(1 + x)).
a(n) = 4*n^3 - 3*n^2 + 3*n - 2 for n>0 and even.
a(n) = 4*n^3 - 3*n^2 + 3*n - 1 for n odd.
(End)

A211619 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and 2w+x+y>2.

Original entry on oeis.org

0, 1, 18, 73, 192, 395, 710, 1157, 1764, 2551, 3546, 4769, 6248, 8003, 10062, 12445, 15180, 18287, 21794, 25721, 30096, 34939, 40278, 46133, 52532, 59495, 67050, 75217, 84024, 93491, 103646, 114509, 126108, 138463, 151602, 165545, 180320, 195947, 212454
Offset: 0

Views

Author

Clark Kimberling, Apr 16 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t = Compile[{{u, _Integer}},
   Module[{s = 0}, (Do[If[2 w + x + y > 2,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 70]]  (* A211619 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
Join[{0, 1},LinearRecurrence[{3, -2, -2, 3, -1},{18, 73, 192, 395, 710},34]] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(1 + 15*x + 21*x^2 + 11*x^3 - 2*x^4 + 2*x^5) / ((1 - x)^4*(1 + x)) + O(x^40))) \\ Colin Barker, Dec 04 2017

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) for n>6.
From Colin Barker, Dec 04 2017: (Start)
G.f.: x*(1 + 15*x + 21*x^2 + 11*x^3 - 2*x^4 + 2*x^5) / ((1 - x)^4*(1 + x)).
a(n) = 4*n^3 - 5*n^2 + 5*n - 4 for n>1 and even.
a(n) = 4*n^3 - 5*n^2 + 5*n - 5 for n>1 and odd.
(End)

A211620 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and -1<=2w+x+y<=1.

Original entry on oeis.org

0, 2, 16, 38, 76, 122, 184, 254, 340, 434, 544, 662, 796, 938, 1096, 1262, 1444, 1634, 1840, 2054, 2284, 2522, 2776, 3038, 3316, 3602, 3904, 4214, 4540, 4874, 5224, 5582, 5956, 6338, 6736, 7142, 7564, 7994, 8440, 8894, 9364, 9842, 10336, 10838, 11356, 11882
Offset: 0

Views

Author

Clark Kimberling, Apr 16 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t = Compile[{{u, _Integer}},
   Module[{s = 0}, (Do[If[-1 <= 2 w + x + y <= 1,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 70]]  (* A211620 *)
%/2                        (* integers *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
Join[{0},LinearRecurrence[{2, 0, -2, 1},{2, 16, 38, 76},42]] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(2*x*(1 + 6*x + 3*x^2 + 2*x^3) / ((1 - x)^3*(1 + x)) + O(x^40))) \\ Colin Barker, Dec 04 2017

Formula

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>4.
From Colin Barker, Dec 04 2017: (Start)
G.f.: 2*x*(1 + 6*x + 3*x^2 + 2*x^3) / ((1 - x)^3*(1 + x)).
a(n) = 6*n^2 - 6*n + 4 for n>0 and even.
a(n) = 6*n^2 - 6*n + 2 for n odd.
(End)
Previous Showing 71-80 of 105 results. Next

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