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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A211621 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+2x+3y>0.

Original entry on oeis.org

0, 3, 29, 103, 247, 484, 843, 1342, 2008, 2866, 3938, 5247, 6822, 8681, 10851, 13357, 16221, 19466, 23121, 27204, 31742, 36760, 42280, 48325, 54924, 62095, 69865, 78259, 87299, 97008, 107415, 118538, 130404, 143038, 156462, 170699, 185778, 201717, 218543
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 + 2 x + 3 y > 0, s = s + 1], {w, #}, {x, #}, {y, #}] &[Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 70]]  (* A211621 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{2, 0, -1, -1, 0, 2, -1},{0, 3, 29, 103, 247, 484, 843},36] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(3 + 23*x + 45*x^2 + 44*x^3 + 22*x^4 + 7*x^5) / ((1 - x)^4*(1 + x)*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Dec 05 2017

Formula

a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7) for n>6.
G.f.: x*(3 + 23*x + 45*x^2 + 44*x^3 + 22*x^4 + 7*x^5) / ((1 - x)^4*(1 + x)*(1 + x + x^2)). - Colin Barker, Dec 05 2017

A211622 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+2x+3y>1.

Original entry on oeis.org

0, 3, 26, 94, 229, 457, 800, 1284, 1931, 2767, 3814, 5098, 6641, 8469, 10604, 13072, 15895, 19099, 22706, 26742, 31229, 36193, 41656, 47644, 54179, 61287, 68990, 77314, 86281, 95917, 106244, 117288, 129071, 141619, 154954, 169102, 184085, 199929, 216656
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 + 2 x + 3 y > 1,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 70]]  (* A211622 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
Join[{0},LinearRecurrence[{3, -2, -2, 3, -1},{3, 26, 94, 229, 457},35]] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(3 + 17*x + 22*x^2 + 5*x^3 + x^4) / ((1 - x)^4*(1 + x)) + O(x^40))) \\ Colin Barker, Dec 05 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 05 2017: (Start)
G.f.: x*(3 + 17*x + 22*x^2 + 5*x^3 + x^4) / ((1 - x)^4*(1 + x)).
a(n) = (8*n^3 - 4*n^2 + 3*n - 2) / 2 for n>0 and even.
a(n) = (16*n^3 - 8*n^2 + 6*n - 2) / 4 for n odd.
(End)

A211623 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and -1<=w+2x+3y<=1.

Original entry on oeis.org

0, 2, 12, 28, 54, 86, 128, 176, 234, 298, 372, 452, 542, 638, 744, 856, 978, 1106, 1244, 1388, 1542, 1702, 1872, 2048, 2234, 2426, 2628, 2836, 3054, 3278, 3512, 3752, 4002, 4258, 4524, 4796, 5078, 5366, 5664, 5968, 6282, 6602, 6932, 7268, 7614, 7966, 8328
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 + 2 x + 3 y <= 1,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 70]]  (* A211623 *)
%/2  (* integers *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
Join[{0},LinearRecurrence[{2, 0, -2, 1},{2, 12, 28, 54},43]] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(2*x*(1 + 4*x + 2*x^2 + x^3) / ((1 - x)^3*(1 + x)) + O(x^40))) \\ Colin Barker, Dec 05 2017

Formula

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>4.
From Colin Barker, Dec 05 2017: (Start)
G.f.: 2*x*(1 + 4*x + 2*x^2 + x^3) / ((1 - x)^3*(1 + x)).
a(n) = 4*n^2 - 3*n + 2 for n>0 and even.
a(n) = 4*n^2 - 3*n + 1 for n odd.
(End)

A211624 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+2x+2y>0.

Original entry on oeis.org

0, 4, 30, 104, 245, 485, 837, 1339, 1998, 2858, 3920, 5234, 6795, 8659, 10815, 13325, 16172, 19424, 23058, 27148, 31665, 36689, 42185, 48239, 54810, 61990, 69732, 78134, 87143, 96863, 107235, 118369, 130200, 142844, 156230, 170480, 185517, 201469, 218253
Offset: 0

Views

Author

Clark Kimberling, Apr 17 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 + 2 x + 2 y > 0,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211624 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1},{0, 4, 30, 104, 245, 485},36] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(4 + 22*x + 40*x^2 + 23*x^3 + 7*x^4) / ((1 - x)^4*(1 + x)^2) + O(x^50))) \\ Colin Barker, Aug 23 2017

Formula

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
From Colin Barker, Aug 23 2017: (Start)
G.f.: x*(4 + 22*x + 40*x^2 + 23*x^3 + 7*x^4) / ((1 - x)^4*(1 + x)^2).
a(n) = (64*n^3 - 14*n^2 + 12*n) / 16 for n even.
a(n) = (64*n^3 - 14*n^2 + 24*n - 10) / 16 for n odd. (End)

A211625 Number of ordered triples (w,x,y) with all terms in {-n,...,-1,1,...,n} and w+3x+3y>0.

Original entry on oeis.org

0, 4, 32, 104, 250, 492, 845, 1349, 2021, 2871, 3949, 5267, 6830, 8698, 10878, 13370, 16244, 19502, 23139, 27235, 31787, 36785, 42319, 48381, 54956, 62144, 69932, 78300, 87358, 97088, 107465, 118609, 130497, 143099, 156545, 170807, 185850, 201814, 218666
Offset: 0

Views

Author

Clark Kimberling, Apr 17 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    Remove["Global`*"];
t = Compile[{{u, _Integer}},
   Module[{s = 0}, (Do[If[w + 3 x + 3 y > 0,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211625 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{2, -1, 2, -4, 2, -1, 2, -1},{0, 4, 32, 104, 250, 492, 845, 1349},36] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(7*x^6+23*x^5+48*x^4+66*x^3+44*x^2+24*x+4)/((x-1)^4*(x^2+x+1)^2) + O(x^100))) \\ Colin Barker, Nov 17 2015

Formula

a(n) = 2*a(n-1)-a(n-2)+2*a(n-3)-4*a(n-4)+2*a(n-5)-a(n-6)+2*a(n-7)-a(n-8).
G.f.: x*(7*x^6+23*x^5+48*x^4+66*x^3+44*x^2+24*x+4) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Nov 17 2015

A211626 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+4x+4y>0.

Original entry on oeis.org

0, 4, 32, 108, 250, 492, 854, 1360, 2021, 2885, 3965, 5285, 6849, 8719, 10901, 13419, 16270, 19530, 23198, 27298, 31820, 36854, 42392, 48458, 55035, 62227, 70019, 78435, 87451, 97185, 107615, 118765, 130604, 143264, 156716, 170984, 186030, 202000, 218858
Offset: 0

Views

Author

Clark Kimberling, Apr 17 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 + 4 x + 4 y > 0,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211626 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{2,-1,0,2,-4,2,0,-1,2,-1},{0,4,32,108,250,492,854,1360,2021,2885},40] (* Harvey P. Dale, Nov 29 2013 *)
  • PARI
    concat(0, Vec(x*(4 + 24*x + 48*x^2 + 66*x^3 + 92*x^4 + 72*x^5 + 48*x^6 + 23*x^7 + 7*x^8) / ((1 - x)^4*(1 + x)^2*(1 + x^2)^2) + O(x^40))) \\ Colin Barker, Dec 05 2017

Formula

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-4) - 4*a(n-5) + 2*a(n-6) - a(n-8) + 2*a(n-9) - a(n-10) for n>9.
G.f.: x*(4 + 24*x + 48*x^2 + 66*x^3 + 92*x^4 + 72*x^5 + 48*x^6 + 23*x^7 + 7*x^8) / ((1 - x)^4*(1 + x)^2*(1 + x^2)^2). - Colin Barker, Dec 05 2017

A211627 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+5x+5y>0.

Original entry on oeis.org

0, 4, 32, 108, 256, 492, 854, 1360, 2034, 2900, 3965, 5285, 6869, 8741, 10925, 13419, 16297, 19559, 23229, 27331, 31854, 36890, 42430, 48498, 55118, 62270, 70064, 78482, 87548, 97286, 107667, 118819, 130715, 143379, 156835, 171045, 186155, 202129, 218991
Offset: 0

Views

Author

Clark Kimberling, Apr 17 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 + 5 x + 5 y > 0,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211627 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{2, -1, 0, 0, 2, -4, 2, 0, 0, -1, 2, -1},{0, 4, 32, 108, 256, 492, 854, 1360, 2034, 2900, 3965, 5285},36] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(4 + 24*x + 48*x^2 + 72*x^3 + 88*x^4 + 118*x^5 + 96*x^6 + 72*x^7 + 48*x^8 + 23*x^9 + 7*x^10) / ((1 - x)^4*(1 + x + x^2 + x^3 + x^4)^2) + O(x^40))) \\ Colin Barker, Dec 05 2017

Formula

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-5) - 4*a(n-6) + 2*a(n-7) - a(n-10) + 2*a(n-11) - a(n-12) for n>11.
G.f.: x*(4 + 24*x + 48*x^2 + 72*x^3 + 88*x^4 + 118*x^5 + 96*x^6 + 72*x^7 + 48*x^8 + 23*x^9 + 7*x^10) / ((1 - x)^4*(1 + x + x^2 + x^3 + x^4)^2). - Colin Barker, Dec 05 2017

A211628 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and 3w+x+y>0.

Original entry on oeis.org

0, 4, 30, 105, 249, 487, 846, 1346, 2012, 2871, 3943, 5253, 6828, 8688, 10858, 13365, 16229, 19475, 23130, 27214, 31752, 36771, 42291, 48337, 54936, 62108, 69878, 78273, 87313, 97023, 107430, 118554, 130420, 143055, 156479, 170717, 185796, 201736, 218562
Offset: 0

Views

Author

Clark Kimberling, Apr 17 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[3 w + x + y > 0,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]   (* A211628 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{3, -3, 2, -3, 3, -1},{0, 4, 30, 105, 249, 487},36] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(4 + 18*x + 27*x^2 + 16*x^3 + 7*x^4) / ((1 - x)^4*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Dec 05 2017

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) for n>5.
G.f.: x*(4 + 18*x + 27*x^2 + 16*x^3 + 7*x^4) / ((1 - x)^4*(1 + x + x^2)). - Colin Barker, Dec 05 2017

A211629 Number of ordered triples (w,x,y) with all terms in {-n, ..., -1, 1, ..., n} and 4w + x + y > 0.

Original entry on oeis.org

0, 4, 31, 105, 252, 492, 851, 1353, 2024, 2884, 3959, 5273, 6852, 8716, 10891, 13401, 16272, 19524, 23183, 27273, 31820, 36844, 42371, 48425, 55032, 62212, 69991, 78393, 87444, 97164, 107579, 118713, 130592, 143236, 156671, 170921, 186012, 201964, 218803
Offset: 0

Views

Author

Clark Kimberling, Apr 17 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[4 w + x + y > 0,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211629 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{3, -3, 1, 1, -3, 3, -1},{0, 4, 31, 105, 252, 492, 851},36] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(4 + 19*x + 24*x^2 + 26*x^3 + 16*x^4 + 7*x^5) / ((1 - x)^4*(1 + x)*(1 + x^2)) + O(x^40))) \\ Colin Barker, Dec 05 2017

Formula

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) for n > 6.
G.f.: x*(4 + 19*x + 24*x^2 + 26*x^3 + 16*x^4 + 7*x^5) / ((1 - x)^4*(1 + x)*(1 + x^2)). - Colin Barker, Dec 05 2017

A211630 Number of ordered triples (w,x,y) with all terms in {-n, ..., -1, 1, ..., n} and 5w + x + y > 0.

Original entry on oeis.org

0, 4, 32, 106, 252, 495, 855, 1359, 2029, 2891, 3970, 5286, 6866, 8732, 10910, 13425, 16297, 19553, 23215, 27309, 31860, 36888, 42420, 48478, 55088, 62275, 70059, 78467, 87521, 97247, 107670, 118810, 130694, 143344, 156786, 171045, 186141, 202101, 218947
Offset: 0

Views

Author

Clark Kimberling, Apr 17 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[5 w + x + y > 0,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211630 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{3, -3, 1, 0, 1, -3, 3, -1},{0, 4, 32, 106, 252, 495, 855, 1359},36] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(4 + 20*x + 22*x^2 + 26*x^3 + 25*x^4 + 16*x^5 + 7*x^6) / ((1 - x)^4*(1 + x + x^2 + x^3 + x^4)) + O(x^40))) \\ Colin Barker, Dec 05 2017

Formula

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) for n > 7.
G.f.: x*(4 + 20*x + 22*x^2 + 26*x^3 + 25*x^4 + 16*x^5 + 7*x^6) / ((1 - x)^4*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Dec 05 2017
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