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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A211522 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w + 5y = 2x.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 6, 8, 11, 13, 16, 19, 23, 27, 31, 35, 40, 45, 51, 56, 62, 68, 75, 82, 89, 96, 104, 112, 121, 129, 138, 147, 157, 167, 177, 187, 198, 209, 221, 232, 244, 256, 269, 282, 295, 308, 322, 336, 351, 365, 380, 395, 411, 427, 443, 459, 476
Offset: 0

Views

Author

Clark Kimberling, Apr 14 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[w - 2 x + 5 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211522 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,1,-1,0,1,-1,-1,1},{0,0,0,1,2,3,4,6},58] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(vector(3), Vec(x^3*(1 + x + x^4) / ((1 - x)^3*(1 + x)*(1 + x + x^2 + x^3 + x^4)) + O(x^40))) \\ Colin Barker, Dec 02 2017

Formula

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-5) - a(n-6) - a(n-7) + a(n-8).
G.f.: x^3*(1 + x + x^4) / ((1 - x)^3*(1 + x)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Dec 02 2017
a(n) ~ 3*n^2/20. - Stefano Spezia, Mar 11 2025

A211524 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w=3x+5y.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 20, 22, 23, 25, 27, 29, 31, 33, 35, 37, 40, 42, 44, 47, 49, 52, 55, 57, 60, 63, 66, 69, 72, 75, 78, 82, 85, 88, 92, 95, 99, 103, 106, 110, 114, 118, 122, 126, 130, 134, 139
Offset: 0

Views

Author

Clark Kimberling, Apr 14 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A008672, A211422.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[-w + 3 x + 5 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211524 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,0,1,-1,1,-1,0,-1,1},{0,0,0,0,0,0,0,0,1},69] (* Ray Chandler, Aug 02 2015 *)

Formula

a(n) = a(n-1)+a(n-3)-a(n-4)+a(n-5)-a(n-6)-a(n-8)+a(n-9).
G.f.: x^8/((1-x)*(1-x^3)*(1-x^5)).

A211533 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w=3x-5y.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 5, 8, 10, 13, 16, 19, 23, 27, 32, 36, 41, 47, 52, 59, 65, 71, 79, 86, 94, 102, 110, 119, 128, 138, 147, 157, 168, 178, 190, 201, 212, 225, 237, 250, 263, 276, 290, 304, 319, 333, 348, 364, 379, 396, 412, 428, 446, 463, 481, 499, 517, 536
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 + 5 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211533 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,0,1,-1,1,-1,0,-1,1},{0,0,1,1,3,4,5,8,10},58] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(vector(2), Vec(x^2*(1 + 2*x^2 + x^4 + x^6) / ((1 - x)^3*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)) + O(x^40))) \\ Colin Barker, Dec 02 2017

Formula

a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) - a(n-6) - a(n-8) + a(n-9).
G.f.: x^2*(1 + 2*x^2 + x^4 + x^6) / ((1 - x)^3*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Dec 02 2017
a(n) ~ n^2/6. - Stefano Spezia, Apr 09 2025

A211535 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w=4x+5y.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 34, 36, 38, 40, 42, 44, 46, 48, 51, 53, 55, 57, 60, 63, 65, 67, 70, 73, 76, 78, 81, 84, 87, 90, 93, 96, 99, 102, 106, 109
Offset: 0

Views

Author

Clark Kimberling, Apr 15 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422, A205772.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[-w + 4 x + 5 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211535 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,0,0,1,0,-1,0,0,-1,1},{0,0,0,0,0,0,0,0,0,1},71] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(vector(9), Vec(x^9 / ((1 - x)^3*(1 + x)*(1 + x^2)*(1 + x + x^2 + x^3 + x^4)) + O(x^100))) \\ Colin Barker, Dec 03 2017

Formula

a(n) = a(n-1) + a(n-4) - a(n-6) - a(n-9) + a(n-10).
G.f.: x^9 / ((1 - x)^3*(1 + x)*(1 + x^2)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Dec 03 2017
a(n)-a(n-1)=A165190(n-9). - R. J. Mathar, Jun 23 2021

A211536 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w=4x-5y.

Original entry on oeis.org

0, 0, 0, 2, 3, 4, 6, 8, 11, 14, 17, 21, 24, 29, 34, 39, 44, 49, 56, 63, 69, 76, 83, 92, 100, 108, 117, 126, 136, 146, 156, 167, 177, 189, 201, 213, 225, 237, 251, 265, 278, 292, 306, 322, 337, 352, 368, 384, 401, 418, 435, 453, 470, 489, 508, 527, 546
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 - 4 x + 5 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211536 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,0,0,1,0,-1,0,0,-1,1},{0,0,0,2,3,4,6,8,11,14},57] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(vector(3), Vec(x^3*(2 + x + x^2 + 2*x^3 + x^6) / ((1 - x)^3*(1 + x)*(1 + x^2)*(1 + x + x^2 + x^3 + x^4)) + O(x^40))) \\ Colin Barker, Dec 03 2017

Formula

a(n) = a(n-1) + a(n-4) - a(n-6) - a(n-9) + a(n-10).
G.f.: x^3*(2 + x + x^2 + 2*x^3 + x^6) / ((1 - x)^3*(1 + x)*(1 + x^2)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Dec 03 2017

A211541 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w=3x-4y.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 6, 7, 11, 11, 17, 18, 24, 25, 33, 34, 43, 44, 54, 56, 67, 68, 81, 83, 96, 98, 113, 115, 131, 133, 150, 153, 171, 173, 193, 196, 216, 219, 241, 244, 267, 270, 294, 298, 323, 326, 353, 357, 384, 388, 417, 421, 451, 455, 486, 491, 523, 527
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[2 w - 3 x + 4 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211541 *)
FindLinearRecurrence[t]

Formula

a(n) = a(n-2)+a(n-3)+a(n-4)-a(n-5)-a(n-6)-a(n-7)+a(n-9).
G.f.: x^2*(1+x+2*x^2+x^3+x^4+x^5+x^6)/((1+x^2)*(1+x+x^2)*(1+x)^2*(1-x)^3). [Bruno Berselli, Jun 15 2012]

A211542 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w=4y-3x.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 10, 14, 17, 22, 26, 32, 36, 44, 49, 57, 63, 73, 79, 90, 97, 109, 117, 130, 138, 153, 162, 177, 187, 204, 214, 232, 243, 262, 274, 294, 306, 328, 341, 363, 377, 401, 415, 440, 455, 481, 497, 524, 540, 569, 586, 615, 633, 664, 682, 714
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[2 w + 3 x - 4 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211542 *)
FindLinearRecurrence[t]
LinearRecurrence[{0,1,1,1,-1,-1,-1,0,1},{0,0,1,2,3,5,8,10,14},57] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(vector(2), Vec(x^2*(1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Dec 03 2017

Formula

a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9).
G.f.: x^2*(1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)). - Colin Barker, Dec 03 2017

A211543 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w=3x+5y.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 14, 16, 18, 21, 23, 25, 28, 31, 34, 37, 40, 43, 47, 51, 54, 58, 62, 66, 71, 75, 79, 84, 89, 94, 99, 104, 109, 115, 121, 126, 132, 138, 144, 151, 157, 163, 170, 177, 184, 191, 198, 205, 213, 221, 228, 236, 244
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[-2 w + 3 x + 5 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]   (* A211543 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,0,1,-1,1,-1,0,-1,1},{0,0,0,0,1,1,1,2,3},63] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(vector(4), Vec(x^4*(1 + x^4) / ((1 - x)^3*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)) + O(x^40))) \\ Colin Barker, Dec 03 2017

Formula

a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) - a(n-6) - a(n-8) + a(n-9).
G.f.: x^4*(1 + x^4) / ((1 - x)^3*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Dec 03 2017

A211544 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w=3x-5y.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 5, 8, 10, 12, 15, 17, 21, 25, 28, 32, 36, 41, 46, 51, 56, 61, 68, 74, 80, 87, 93, 101, 109, 116, 124, 132, 141, 150, 159, 168, 177, 188, 198, 208, 219, 229, 241, 253, 264, 276, 288, 301, 314, 327, 340, 353, 368, 382, 396, 411, 425, 441
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[2 w - 3 x + 5 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]   (* A211544 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,0,1,-1,1,-1,0,-1,1},{0,0,0,0,1,1,1,2,3},63] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(vector(3), Vec(x^3*(1 + x)*(1 + x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)) + O(x^40))) \\ Colin Barker, Dec 03 2017

Formula

a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) - a(n-6) - a(n-8) + a(n-9).
G.f.: x^3*(1 + x)*(1 + x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)) - Colin Barker, Dec 03 2017

A211545 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, 29, 99, 238, 470, 819, 1309, 1964, 2808, 3865, 5159, 6714, 8554, 10703, 13185, 16024, 19244, 22869, 26923, 31430, 36414, 41899, 47909, 54468, 61600, 69329, 77679, 86674, 96338, 106695, 117769, 129584, 142164, 155533, 169715, 184734, 200614, 217379
Offset: 0

Views

Author

Clark Kimberling, Apr 16 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Examples

			a(1) counts these triples: (-1,1,1), (1,-1,1), (1,1,-1), (1,1,1).

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]]  (* A211545 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,4,29,99},36] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(4 + 13*x + 7*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 + 13*x + 7*x^2) / (1 - x)^4.
a(n) = (n*(3 - 3*n + 8*n^2))/2.
(End)
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