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.

Previous Showing 41-50 of 76 results. Next

A213395 Number of (w,x,y) with all terms in {0,...,n} and max(|w-x|,|x-y|) = w.

Original entry on oeis.org

1, 4, 11, 19, 31, 44, 62, 79, 103, 125, 154, 181, 216, 247, 288, 324, 370, 411, 463, 508, 566, 616, 679, 734, 803, 862, 937, 1001, 1081, 1150, 1236, 1309, 1401, 1479, 1576, 1659, 1762, 1849, 1958, 2050, 2164, 2261, 2381, 2482, 2608, 2714, 2845
Offset: 0

Views

Author

Clark Kimberling, Jun 12 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 == Max[Abs[w - x], Abs[x - y]], s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A213395 *)
  • PARI
    a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 1,0,-2,-1,1,2,0]^n*[1;4;11;19;31;44;62])[1,1] \\ Charles R Greathouse IV, Nov 27 2016

Formula

a(n) = 2*a(n-2) +a(n-3) -a(n-4) -2*a(n-5) +a(n-7).
G.f.: (1 +4*x +9*x^2 +10*x^3 +6*x^4 +x^5) / ((1-x)^3*(1+x)^2*(1+x+x^2)).

A213398 Number of (w,x,y) with all terms in {0,...,n} and min(|w-x|,|x-y|) = x.

Original entry on oeis.org

1, 4, 10, 17, 27, 38, 52, 67, 85, 104, 126, 149, 175, 202, 232, 263, 297, 332, 370, 409, 451, 494, 540, 587, 637, 688, 742, 797, 855, 914, 976, 1039, 1105, 1172, 1242, 1313, 1387, 1462, 1540, 1619, 1701, 1784, 1870, 1957, 2047, 2138, 2232, 2327
Offset: 0

Views

Author

Clark Kimberling, Jun 12 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[x == Min[Abs[w - x], Abs[x - y]], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A213398 *)
LinearRecurrence[{2,0,-2,1},{1,4,10,17},50] (* Harvey P. Dale, Aug 05 2019 *)
  • PARI
    first(n) = Vec((1 + 2*x + 2*x^2 - x^3)/((1 - x)^3*(1 + x)) + O(x^n)) \\ Iain Fox, Feb 01 2018

Formula

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1 + 2*x + 2*x^2 - x^3)/((1 - x)^3*(1 + x)).
a(n) = (n+1)^2 + floor(n/2). [Wesley Ivan Hurt, Jul 15 2013]
From Iain Fox, Feb 01 2018: (Start)
E.g.f.: (1 + e^(2*x) * (3 + 14*x + 4*x^2))/(4 * e^x).
a(n) = (4*n^2 + 10*n + (-1)^n + 3)/4.
(End)

A213480 Number of (w,x,y) with all terms in {0,...,n} and |w-x| + |x-y| != w+x+y.

Original entry on oeis.org

0, 4, 16, 46, 95, 175, 285, 439, 634, 886, 1190, 1564, 2001, 2521, 3115, 3805, 4580, 5464, 6444, 7546, 8755, 10099, 11561, 13171, 14910, 16810, 18850, 21064, 23429, 25981, 28695, 31609, 34696, 37996, 41480, 45190, 49095, 53239, 57589
Offset: 0

Views

Author

Clark Kimberling, Jun 13 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 != Abs[w - x] + Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A213480 *)

Formula

a(n) + A213479(n) = (n+1)^3.
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.: (4*x + 8*x^2 + 10*x^3 + 3*x^4 - x^5)/((1 - x)^4*(1 + x)^2).
a(n) = (8*n^3 + 15*n^2 + 4*n + 5 - (2*n+5)*((n+1) mod 2))/8. - Ayoub Saber Rguez, Nov 20 2021

A213481 Number of triples (w,x,y) with all terms in {0,...,n} and |w-x| + |x-y| <= w+x+y.

Original entry on oeis.org

1, 7, 25, 59, 117, 202, 323, 482, 689, 945, 1261, 1637, 2085, 2604, 3207, 3892, 4673, 5547, 6529, 7615, 8821, 10142, 11595, 13174, 14897, 16757, 18773, 20937, 23269, 25760, 28431, 31272, 34305, 37519, 40937, 44547, 48373, 52402, 56659
Offset: 0

Views

Author

Clark Kimberling, Jun 13 2012

Keywords

Comments

For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A212959, A213479, A213482.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x + y >= Abs[w - x] + Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A213481 *)

Formula

a(n) + A006918(n) = (n+1)^3.
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 + 6*x^3 + x^4)/((1 - x)^4*(1 + x)^2).
From Ayoub Saber Rguez, Dec 29 2021: (Start)
a(n) = A213482(n) + A213479(n).
a(n) = (23*n^3 + 66*n^2 + 64*n + 24 - (3*n+6)*(n mod 2))/24. (End)

A213483 Number of (w,x,y) with all terms in {0,...,n} and |w-x| + |x-y| >= w+x+y.

Original entry on oeis.org

1, 5, 13, 23, 38, 55, 78, 103, 135, 169, 211, 255, 308, 363, 428, 495, 573, 653, 745, 839, 946, 1055, 1178, 1303, 1443, 1585, 1743, 1903, 2080, 2259, 2456, 2655, 2873, 3093, 3333, 3575, 3838, 4103, 4390, 4679, 4991, 5305, 5643, 5983, 6348
Offset: 0

Views

Author

Clark Kimberling, Jun 13 2012

Keywords

Comments

a(n) + A213482(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[w + x + y <= Abs[w - x] + Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A213483 *)
LinearRecurrence[{2,1,-4,1,2,-1},{1,5,13,23,38,55},50] (* Harvey P. Dale, Sep 11 2019 *)

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 + 3*x + 2*x^2 - 4*x^3 - 2*x^4 + x^5)/((1 - x)^4*(1 + x)^2).
From Ayoub Saber Rguez, Dec 31 2021: (Start)
a(n) + A213482(n) = (n+1)^3.
a(n) = A213479(n) + A006918(n).
a(n)= (n^3 + 33*n^2 + 71*n + 15 + (3*n+9)*((n+1) mod 2))/24. (End)

A213485 Number of (w,x,y) with all terms in {0,...,n} and |w-x|+|x-y|+|y-w| != w+x+y.

Original entry on oeis.org

0, 4, 20, 54, 109, 191, 309, 469, 674, 930, 1246, 1628, 2079, 2605, 3215, 3915, 4708, 5600, 6600, 7714, 8945, 10299, 11785, 13409, 15174, 17086, 19154, 21384, 23779, 26345, 29091, 32023, 35144, 38460, 41980, 45710, 49653, 53815, 58205
Offset: 0

Views

Author

Clark Kimberling, Jun 13 2012

Keywords

Comments

a(n) + A213484 = (n+1)^3.
For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A212959, A213484.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x + y != Abs[w - x] + Abs[x - y] + Abs[y - w], s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A213485 *)

Formula

a(n) = 4*a(n-1)-7*a(n-2)+8*a(n-3)-7*a(n-4)+4*a(n-5)-a(n-6).
G.f.: (x (4 + 4*x + 2*x^2 + x^3 + x^4))/((1 - x)^4 (1 + x^2)).

A213486 Number of (w,x,y) with all terms in {0,...,n} and |w-x|+|x-y|+|y-w| > w+x+y.

Original entry on oeis.org

0, 3, 12, 27, 48, 78, 120, 174, 240, 321, 420, 537, 672, 828, 1008, 1212, 1440, 1695, 1980, 2295, 2640, 3018, 3432, 3882, 4368, 4893, 5460, 6069, 6720, 7416, 8160, 8952, 9792, 10683, 11628, 12627, 13680, 14790, 15960, 17190, 18480, 19833
Offset: 0

Views

Author

Clark Kimberling, Jun 13 2012

Keywords

Comments

Every term is divisible by 3.
For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A212959, A213487.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[w + x + y < Abs[w - x] + Abs[x - y] + Abs[y - w], s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A213486 *)

Formula

a(n) = 4*a(n-1)-7*a(n-2)+8*a(n-3)-7*a(n-4)+4*a(n-5)-a(n-6).
G.f.: 3*x/((1 - x)^4 (1 + x^2)).
a(n) = (n+1)^3 - A213487(n).

A213487 Number of (w,x,y) with all terms in {0,...,n} and |w-x|+|x-y|+|y-w| <= w+x+y.

Original entry on oeis.org

1, 5, 15, 37, 77, 138, 223, 338, 489, 679, 911, 1191, 1525, 1916, 2367, 2884, 3473, 4137, 4879, 5705, 6621, 7630, 8735, 9942, 11257, 12683, 14223, 15883, 17669, 19584, 21631, 23816, 26145, 28621, 31247, 34029, 36973, 40082, 43359, 46810
Offset: 0

Views

Author

Clark Kimberling, Jun 13 2012

Keywords

Comments

For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A212959, A213486.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[w + x + y >= Abs[w - x] + Abs[x - y] + Abs[y - w], s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A213487 *)
LinearRecurrence[{4,-7,8,-7,4,-1},{1,5,15,37,77,138},50] (* Harvey P. Dale, Jun 19 2024 *)

Formula

a(n) = 4*a(n-1)-7*a(n-2)+8*a(n-3)-7*a(n-4)+4*a(n-5)-a(n-6).
G.f.: (1 + x + 2*x^2 + 4*x^3 + x^4)/((1 - x)^4 (1 + x^2)).
a(n) = (n+1)^3 - A213486(n).

A213489 Number of (w,x,y) with all terms in {0,...,n} and |w-x| + |x-y| + |y-w| >= w + x + y.

Original entry on oeis.org

1, 7, 19, 37, 64, 103, 154, 217, 295, 391, 505, 637, 790, 967, 1168, 1393, 1645, 1927, 2239, 2581, 2956, 3367, 3814, 4297, 4819, 5383, 5989, 6637, 7330, 8071, 8860, 9697, 10585, 11527, 12523, 13573, 14680, 15847, 17074, 18361, 19711, 21127
Offset: 0

Views

Author

Clark Kimberling, Jun 13 2012

Keywords

Comments

For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A212959, A213488.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x + y <= Abs[w - x] + Abs[x - y] + Abs[y - w],
s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A213489 *)

Formula

a(n) = 4*a(n-1) - 7*a(n-2) + 8*a(n-3) - 7*a(n-4) + 4*a(n-5) - a(n-6).
G.f.: (1 + 3*x - 2*x^2 + 2* x^3 - x^5)/((1 - x)^4 (1 + x^2)). [corrected by Georg Fischer, May 10 2019]
a(n) + A213488(n) = (n+1)^3.

A213498 Number of (w,x,y) with all terms in {0,...,n} and w != max(|w-x|,|x-y|,|y-w|).

Original entry on oeis.org

0, 4, 15, 43, 88, 164, 267, 415, 600, 844, 1135, 1499, 1920, 2428, 3003, 3679, 4432, 5300, 6255, 7339, 8520, 9844, 11275, 12863, 14568, 16444, 18447, 20635, 22960, 25484, 28155, 31039, 34080, 37348, 40783, 44459, 48312, 52420, 56715
Offset: 0

Views

Author

Clark Kimberling, Jun 14 2012

Keywords

Comments

a(n)+A212965 = (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[w != Max[Abs[w - x], Abs[x - y], Abs[y - w]], s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A213498 *)
LinearRecurrence[{2,1,-4,1,2,-1},{0,4,15,43,88,164},50] (* Harvey P. Dale, Mar 27 2020 *)

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.: (4*x + 7*x^2 + 9*x^3 + 3*x^4 + x^5)/((-1 + x)^4*(1 + x)^2).
a(n) = (2*n*(4*n^2+5*n+5) - (2*n+1)*(-1)^n + 1)/8.
Previous Showing 41-50 of 76 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.