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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A212976 Number of (w,x,y) with all terms in {0,...,n} and odd range.

Original entry on oeis.org

0, 6, 12, 36, 60, 114, 168, 264, 360, 510, 660, 876, 1092, 1386, 1680, 2064, 2448, 2934, 3420, 4020, 4620, 5346, 6072, 6936, 7800, 8814, 9828, 11004, 12180, 13530, 14880, 16416, 17952, 19686, 21420, 23364, 25308, 27474, 29640, 32040
Offset: 0

Views

Author

Clark Kimberling, Jun 03 2012

Keywords

Comments

a(n) + A212975(n) = (n+1)^3. Six divides every term.
For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A005993, A212959, A212975.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Mod[Max[w, x, y] - Min[w, x, y], 2] == 1,
   s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212976 *)
m/6  (* A005993 except for initial 0 *)
LinearRecurrence[{2,1,-4,1,2,-1},{0,6,12,36,60,114},40] (* Harvey P. Dale, Jan 21 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).
G.f.: f(x)/g(x), where f(x) = 6*x*(1 + x^2) and g(x) = ((1-x)^4)*(1+x)^2.
a(n+1) = 6*A005993(n). [Bruno Berselli, Jun 15 2012]

A212978 Number of (w,x,y) with all terms in {0,...,n} and range = 2*n-w-x.

Original entry on oeis.org

1, 5, 11, 20, 32, 46, 63, 83, 105, 130, 158, 188, 221, 257, 295, 336, 380, 426, 475, 527, 581, 638, 698, 760, 825, 893, 963, 1036, 1112, 1190, 1271, 1355, 1441, 1530, 1622, 1716, 1813, 1913, 2015, 2120, 2228, 2338, 2451, 2567, 2685, 2806, 2930
Offset: 0

Views

Author

Clark Kimberling, Jun 03 2012

Keywords

Comments

For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A212959.
Second bisection of A281333.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Max[w, x, y] - Min[w, x, y] == 2 n - w - x,
  s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212978 *)
LinearRecurrence[{2,-1,1,-2,1},{1,5,11,20,32},50] (* Harvey P. Dale, Sep 30 2017 *)

Formula

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
G.f.: (1 + 3*x + 2*x^2 + 2*x^3)/((1 - x)^3*(1 + x + x^2)). [corrected by Bruno Berselli, Jan 23 2017]

A212979 Number of (w,x,y) with all terms in {0,...,n} and range=average.

Original entry on oeis.org

1, 1, 1, 7, 10, 13, 19, 25, 34, 40, 49, 61, 70, 82, 94, 109, 124, 136, 154, 172, 190, 208, 226, 250, 271, 292, 316, 340, 367, 391, 418, 448, 475, 505, 535, 568, 601, 631, 667, 703, 739, 775, 811, 853, 892, 931, 973, 1015, 1060, 1102, 1147, 1195, 1240
Offset: 0

Views

Author

Clark Kimberling, Jun 03 2012

Keywords

Comments

For a guide to related sequences, see A212959.

Examples

			a(3)=7 counts these (w,x,y): (0,0,0) and the six permutations of (1,2,3).
G.f. = 1 + x + x^2 + 7*x^3 + 10*x^4 + 13*x^5 + 19*x^6 + 25*x^7 + 34*x^8 + ... - _Michael Somos_, Jan 25 2024

Links

Crossrefs

Cf. A212959, A368482.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Max[w, x, y] - Min[w, x, y] == (w + x + y)/3, s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212979 *)
a[ n_] := If[n<0, a[-1-n], Sum[ Boole[Max[t] - Min[t] == Mean[t]], {t, Tuples[Range[0, n], 3]}]]; (* Michael Somos, Jan 25 2024 *)
a[ n_] := (9*(n^2+n) + 6*{10, 7, 1, 12, 10, 5, 7, 6, 12, 5}[[1 + Min[Mod[n, 20], Mod[-1-n, 20]]]])/20 - 2; (* Michael Somos, Jan 25 2024 *)
  • PARI
    {a(n) = (9*(n^2+n) + 6*[10, 7, 1, 12, 10, 5, 7, 6, 12, 5][1 + min(n%20, (-1-n)%20)])/20 - 2}; /* Michael Somos, Jan 25 2024 */

Formula

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) - 2*a(n-6) + 2*a(n-7) - 2*a(n-8) + a(n-9).
G.f.: (1 - x + x^2 + 5*x^3 - 3*x^4 + 5*x^5 + x^6 - x^7 + x^8)/(1 - 2*x + 2*x^2 - 2*x^3 + x^4 - x^5 + 2*x^6 - 2*x^7 + 2*x^8 - x^9).
From Michael Somos, Jan 25 2024: (Start)
a(n) = a(-1-n) for all n in Z.
G.f.: (1 + x)*(1 - x + x^2 + 5*x^3 - 3*x^4 + 5*x^5 + x^6 - x^7 + x^8)/((1 - x)*(1 - x^4)*(1 - x^5)). (End)
For n > 2, a(n) = 3 * A368482(n+3) + 4. - Helmut Ruhland, Jan 31 2024

A212981 Number of (w,x,y) with all terms in {0,...,n} and w <= x + y and x < y.

Original entry on oeis.org

0, 2, 8, 21, 43, 77, 125, 190, 274, 380, 510, 667, 853, 1071, 1323, 1612, 1940, 2310, 2724, 3185, 3695, 4257, 4873, 5546, 6278, 7072, 7930, 8855, 9849, 10915, 12055, 13272, 14568, 15946, 17408, 18957, 20595, 22325, 24149, 26070, 28090
Offset: 0

Views

Author

Clark Kimberling, Jun 04 2012

Keywords

Comments

For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A212959.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w <= x + y && x < y, s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212981 *)
LinearRecurrence[{3,-2,-2,3,-1},{0,2,8,21,43},50] (* Harvey P. Dale, Jul 31 2013 *)

Formula

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5).
G.f.: f(x)/g(x), where f(x)=2*x + 2*x^2 + x^3 and g(x)=(1+x)*(1-x)^4.
a(n) = (20*n^3+42*n^2+28*n+3*(1-(-1)^n))/48. - Luce ETIENNE, Feb 17 2015

A212982 Number of (w,x,y) with all terms in {0,...,n} and w

Original entry on oeis.org

0, 3, 11, 27, 53, 92, 146, 218, 310, 425, 565, 733, 931, 1162, 1428, 1732, 2076, 2463, 2895, 3375, 3905, 4488, 5126, 5822, 6578, 7397, 8281, 9233, 10255, 11350, 12520, 13768, 15096, 16507, 18003, 19587, 21261, 23028, 24890, 26850, 28910, 31073, 33341, 35717
Offset: 0

Views

Author

Clark Kimberling, Jun 04 2012

Keywords

Comments

For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A212959.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < x + y && x <= y, s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212982 *)
  • PARI
    concat(0, Vec(x*(3+2*x)/((1-x)^4*(1+x)) + O(x^100))) \\ Colin Barker, Jan 28 2016

Formula

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5).
G.f.: f(x)/g(x), where f(x)=3*x + 2*x^2 and g(x)=(1+x)*(1-x)^4.
From Colin Barker, Jan 28 2016: (Start)
a(n) = (20*n^3+66*n^2+52*n-3*(-1)^n+3)/48.
a(n) = (10*n^3+33*n^2+26*n)/24 for n even.
a(n) = (10*n^3+33*n^2+26*n+3)/24 for n odd.
(End)

A212983 Number of (w,x,y) with all terms in {0,...,n} and w<=x+y and x<=y.

Original entry on oeis.org

1, 5, 15, 33, 62, 104, 162, 238, 335, 455, 601, 775, 980, 1218, 1492, 1804, 2157, 2553, 2995, 3485, 4026, 4620, 5270, 5978, 6747, 7579, 8477, 9443, 10480, 11590, 12776, 14040, 15385, 16813, 18327, 19929, 21622, 23408, 25290, 27270, 29351
Offset: 0

Views

Author

Clark Kimberling, Jun 04 2012

Keywords

Comments

For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A002620, A212959, A212982.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w <= x + y && x <= y, s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212983 *)

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
G.f.: (1 + 2*x + 2*x^2)/((1 + x)*(1 - x)^4).
From Ayoub Saber Rguez, Oct 11 2021: (Start)
a(n) = A212982(n) + A002620(n+2).
a(n) = (10*n^3 + 39*n^2 + 50*n + 24 - 3*(n mod 2))/24. (End)

A212986 Number of (w,x,y) with all terms in {0,...,n} and 2w = 3x+y.

Original entry on oeis.org

1, 1, 3, 5, 7, 10, 14, 17, 22, 27, 32, 38, 45, 51, 59, 67, 75, 84, 94, 103, 114, 125, 136, 148, 161, 173, 187, 201, 215, 230, 246, 261, 278, 295, 312, 330, 349, 367, 387, 407, 427, 448, 470, 491, 514, 537, 560, 584, 609, 633, 659, 685, 711, 738, 766
Offset: 0

Views

Author

Clark Kimberling, Jun 04 2012

Keywords

Comments

For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A212959.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[2 w == 3 x + y, s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 70]]   (* A212986 *)

Formula

a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6).
G.f.: f(x)/g(x), where f(x) = 1 + x^2 + x^3 and g(x) = (1 + 2*x + 2*x^2 + x^3)*((1-x)^3).

A213390 Number of (w,x,y) with all terms in {0,...,n} and max(w,x,y) >= 2*min(w,x,y).

Original entry on oeis.org

1, 7, 25, 55, 109, 181, 289, 421, 601, 811, 1081, 1387, 1765, 2185, 2689, 3241, 3889, 4591, 5401, 6271, 7261, 8317, 9505, 10765, 12169, 13651, 15289, 17011, 18901, 20881, 23041, 25297, 27745, 30295, 33049, 35911, 38989, 42181, 45601
Offset: 0

Views

Author

Clark Kimberling, Jun 11 2012

Keywords

Comments

a(n)+A213389(n) = (n+1)^3.
For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A212959.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Max[w, x, y] >= 2*Min[w, x, y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 50]]   (* A213390 *)

Formula

a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6).
G.f.: (1 + 5*x + 10*x^2 + 2*x^3 + x^4 - x^5)/((1 - x)^4*(1 + x)^2).
a(n) = (6*n^3+24*n^2+21*n+8+3*n*(-1)^n)/8. - Luce ETIENNE, Jul 17 2016

A213392 Number of (w,x,y) with all terms in {0,...,n} and 2*max(w,x,y) >= 3*min(w,x,y).

Original entry on oeis.org

1, 7, 25, 61, 115, 199, 319, 469, 667, 919, 1213, 1573, 2005, 2491, 3061, 3721, 4447, 5275, 6211, 7225, 8359, 9619, 10969, 12457, 14089, 15823, 17713, 19765, 21931, 24271, 26791, 29437, 32275, 35311, 38485, 41869, 45469, 49219, 53197
Offset: 0

Views

Author

Clark Kimberling, Jun 11 2012

Keywords

Comments

For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A212959, A213391.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2*Max[w, x, y] >= 3*Min[w, x, y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (* A213391 *)

Formula

a(n) + A213391(n) = (n+1)^3.
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.: -(-1 - 5*x - x^6 - 12*x^2 - 16*x^3 - 8*x^4 - 6*x^5 + x^7) / ((x^2 + x + 1)^2*(x-1)^4).
From Ayoub Saber Rguez, Feb 01 2022: (Start)
a(n) = A213393(n) + A092076(n).
a(n) = (8*n^3 + 27*n^2 + 21*n + 6*n*(((n+1) mod 3) mod 2) + 7 + 2*((2*n+1) mod 3))/9. (End)
From Jon E. Schoenfield, Feb 02 2022: (Start)
a(n) = (8*n^3 + 27*n^2 + 27*n + 9)/9 if n == 0 (mod 3);
= (8*n^3 + 27*n^2 + 21*n + 7)/9 if n == 1 (mod 3);
= (8*n^3 + 27*n^2 + 21*n + 11)/9 if n == 2 (mod 3).
(End)

A213393 Number of (w,x,y) with all terms in {0,...,n} and 2*max(w,x,y) > 3*min(w,x,y).

Original entry on oeis.org

0, 6, 24, 54, 108, 192, 300, 450, 648, 882, 1176, 1536, 1944, 2430, 3000, 3630, 4356, 5184, 6084, 7098, 8232, 9450, 10800, 12288, 13872, 15606, 17496, 19494, 21660, 24000, 26460, 29106, 31944, 34914, 38088, 41472, 45000, 48750, 52728
Offset: 0

Views

Author

Clark Kimberling, Jun 11 2012

Keywords

Comments

Every term is even.
For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A212959, A213391.
Cf. A092076, A190798, A213392.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2*Max[w, x, y] > 3*Min[w, x, y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (* A213393 *)
m/2   (* integers *)

Formula

a(n) + A213391(n+1) = (n+1)^3.
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.: 6*x*(x^2+1)*(x+1)^2 / ((x^2+x+1)^2*(x-1)^4).
From Ayoub Saber Rguez, Feb 01 2022: (Start)
a(n) = 6*A190798(n+1).
a(n) = A213392(n) - A092076(n).
a(n) = (8*n^2+16*n+8-8*n*((2*n+2) mod 3)-8*((2*n+2) mod 3)+2*((2*n+2) mod 3)^2)/3. (End)
E.g.f.: 2*exp(-x/2)*(6*exp(3*x/2)*(1 + x*(13 + 2*x*(6 + x))) - 3*(2 + x)*cos(sqrt(3)*x/2) - sqrt(3)*(2 - 3*x)*sin(sqrt(3)*x/2))/27. - Stefano Spezia, Feb 25 2023
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