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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A212684 Number of (w,x,y,z) with all terms in {1,...,n} and |x-y|=n-w+|y-z|.

Original entry on oeis.org

0, 1, 6, 19, 42, 79, 132, 205, 300, 421, 570, 751, 966, 1219, 1512, 1849, 2232, 2665, 3150, 3691, 4290, 4951, 5676, 6469, 7332, 8269, 9282, 10375, 11550, 12811, 14160, 15601, 17136, 18769, 20502, 22339, 24282, 26335, 28500, 30781, 33180
Offset: 0

Views

Author

Clark Kimberling, May 24 2012

Keywords

Comments

For a guide to related sequences, see A211795.
Also the number of (w,x,y) with all terms in {0,...,n-1} and |w-x|>=|x-y|, see A212959. Clark Kimberling, Jun 02 2012

Links

Crossrefs

Cf. A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[x - y] == n - w + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212684 *)
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 6, 19, 42}, 41] (* Bruno Berselli, Jun 07 2012 *)
  • Maxima
    makelist(coeff(taylor(x*(1+3*x+3*x^2-x^3)/((1+x)*(1-x)^4), x, 0, n), x, n), n, 0, 40); /* Bruno Berselli, May 07 2012 */

Formula

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5).
a(n) + A212683(n) = n^3. Clark Kimberling, Jun 02 2012
G.f.: x*(1+3*x+3*x^2-x^3)/((1+x)*(1-x)^4). [Bruno Berselli, Jun 07 2012]
a(n) = (2*n*(n+2)*(2*n-1)-(-1)^n+1)/8. [Bruno Berselli, Jun 07 2012]

A212689 Number of (w,x,y,z) with all terms in {1,...,n} and 2|w-x|>n+|y-z|.

Original entry on oeis.org

0, 0, 0, 6, 20, 58, 124, 244, 424, 700, 1080, 1610, 2300, 3206, 4340, 5768, 7504, 9624, 12144, 15150, 18660, 22770, 27500, 32956, 39160, 46228, 54184, 63154, 73164, 84350, 96740, 110480, 125600, 142256, 160480, 180438, 202164, 225834
Offset: 0

Views

Author

Clark Kimberling, May 25 2012

Keywords

Comments

a(n)+A212690(n)=n^4.
For a guide to related sequences, see A211795.

Links

Crossrefs

Cf. A211795, A212690.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 Abs[w - x] > n + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212689 *)
%/2 (* integers *)
LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {0, 0, 0, 6, 20, 58, 124}, 40]

Formula

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

A212740 Number of (w,x,y,z) with all terms in {0,...,n} and max{w,x,y,z}<2*min{w,x,y,z}.

Original entry on oeis.org

0, 1, 2, 17, 32, 97, 162, 337, 512, 881, 1250, 1921, 2592, 3697, 4802, 6497, 8192, 10657, 13122, 16561, 20000, 24641, 29282, 35377, 41472, 49297, 57122, 66977, 76832, 89041, 101250, 116161, 131072, 149057, 167042, 188497, 209952
Offset: 0

Views

Author

Clark Kimberling, May 26 2012

Keywords

Comments

a(n)+A212741(n)=n^4. For a guide to related sequences, see A211795.

Links

Crossrefs

Cf. A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Max[w, x, y, z] < 2 Min[w, x, y, z], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212740 *)

Formula

a(n) = 2*a(n-1)+2*a(n-2)-6*a(n-3)+6*a(n-5)-2*a(n-6)-2*a(n-7)+a(n-8).
G.f.: -x*(1+x^2)*(x^4+10*x^2+1) / ( (1+x)^3*(x-1)^5 ).
a(n) = 3*n^2/8 +1/16 +n^4/8 -(-1)^n/16 -3*(-1)^n*n^2/8. - R. J. Mathar, Jul 01 2013

A212743 Number of (w,x,y,z) with all terms in {0,...,n} and max{w,x,y,z}>2*min{w,x,y,z}.

Original entry on oeis.org

0, 14, 64, 224, 528, 1134, 2064, 3584, 5680, 8750, 12720, 18144, 24864, 33614, 44128, 57344, 72864, 91854, 113760, 140000, 169840, 204974, 244464, 290304, 341328, 399854, 464464, 537824, 618240, 708750, 807360, 917504, 1036864
Offset: 0

Views

Author

Clark Kimberling, May 26 2012

Keywords

Comments

Also the number of (w,x,y,z) with all terms in {0,...,n} and at least one term < range{w,x,y,z}.
Every term is even.
a(n)+A212742(n)=n^4.
For a guide to related sequences, see A211795.

Crossrefs

Cf. A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Max[w, x, y, z] > 2 Min[w, x, y, z], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212743 *)
%/2 (* integers *)

Formula

a(n)=2*a(n-1)+2*a(n-2)-6*a(n-3)+6*a(n-5)-2*a(n-6)-2*a(n-7)+a(n-8).
G.f.: f(x)/g(x), where f(x)=-7*x-18*x^2-34*x^3-18*x^4-7*x^5 and g(x)=((1-x)^5)*(1+x)^3.

A211787 Number of (w,x,y,z) with all terms in {1,...,n} and w*x<=2*y*z.

Original entry on oeis.org

0, 1, 15, 66, 201, 469, 958, 1735, 2937, 4656, 7050, 10242, 14461, 19813, 26569, 34904, 45086, 57293, 71898, 89050, 109201, 132534, 159424, 190167, 225296, 264966, 309685, 359823, 415889, 478142, 547302, 623514, 707593, 799821
Offset: 0

Views

Author

Clark Kimberling, Apr 27 2012

Keywords

Comments

a(n)+A211797(n)=n^4.
See A211795 for a guide to related sequences.

Links

Crossrefs

Cf. A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w*x <= 2 y*z, s = s + 1],
    {w, 1, #}, {x, 1, #}, {y, 1, #},
      {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211787 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A211797 Number of 4-tuples (w,x,y,z) with all terms in {1,...,n} and w*x>2*y*z.

Original entry on oeis.org

0, 0, 1, 15, 55, 156, 338, 666, 1159, 1905, 2950, 4399, 6275, 8748, 11847, 15721, 20450, 26228, 33078, 41271, 50799, 61947, 74832, 89674, 106480, 125659, 147291, 171618, 198767, 229139, 262698, 300007, 340983, 386100, 435544, 489598
Offset: 0

Views

Author

Clark Kimberling, Apr 27 2012

Keywords

Comments

a(n)+A211787(n)=n^4.
See A211795 for a guide to related sequences.

Links

Crossrefs

Cf. A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w*x > 2 y*z, s = s + 1], {w, 1, #},
      {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211797 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A211918 Number of (w,x,y,z) with all terms in {1,...,n} and w*x>3*y*z.

Original entry on oeis.org

0, 0, 1, 6, 32, 100, 219, 441, 797, 1312, 2061, 3107, 4451, 6248, 8526, 11336, 14823, 19090, 24122, 30164, 37253, 45501, 55091, 66154, 78663, 92979, 109170, 127281, 147629, 170403, 195536, 223509, 254329, 288192, 325385, 366061
Offset: 0

Views

Author

Clark Kimberling, Apr 27 2012

Keywords

Comments

a(n)+A211912(n)=n^4.
See A211795 for a guide to related sequences.

Crossrefs

Cf. A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*x > 3 y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211918 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A211919 Number of (w,x,y,z) with all terms in {1,...,n} and w*x>=3*y*z.

Original entry on oeis.org

0, 0, 1, 14, 46, 118, 267, 497, 871, 1438, 2215, 3273, 4713, 6526, 8844, 11758, 15303, 19590, 24796, 30862, 38053, 46457, 56115, 67206, 79971, 94357, 110628, 128979, 149481, 172291, 197776, 225789, 256763, 290878, 328179, 369011
Offset: 0

Views

Author

Clark Kimberling, Apr 28 2012

Keywords

Comments

a(n)+A211920(n)=n^4.
See A211795 for a guide to related sequences.

Crossrefs

Cf. A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*x >= 3 y*z, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211919 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A211921 Number of (w,x,y,z) with all terms in {1,...,n} and 2w*x<=3*y*z.

Original entry on oeis.org

0, 1, 11, 58, 176, 407, 840, 1536, 2591, 4133, 6268, 9119, 12895, 17684, 23706, 31201, 40315, 51239, 64352, 79770, 97829, 118795, 142947, 170548, 202114, 237757, 277915, 323080, 373489, 429449, 491670, 560274, 635851, 718858, 809692
Offset: 0

Views

Author

Clark Kimberling, Apr 28 2012

Keywords

Comments

a(n)+A211922(n)=n^4.
See A211795 for a guide to related sequences.

Crossrefs

Cf. A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 w*x <= 3 y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211921 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A211922 Number of (w,x,y,z) with all terms in {1,...,n} and 2*w*x>3*y*z.

Original entry on oeis.org

0, 0, 5, 23, 80, 218, 456, 865, 1505, 2428, 3732, 5522, 7841, 10877, 14710, 19424, 25221, 32282, 40624, 50551, 62171, 75686, 91309, 109293, 129662, 152868, 179061, 208361, 241167, 277832, 318330, 363247, 412725, 467063, 526644, 591736, 662511, 739687, 823382
Offset: 0

Views

Author

Clark Kimberling, Apr 28 2012

Keywords

Comments

a(n)+A211921(n)=n^4.
See A211795 for a guide to related sequences.

Crossrefs

Cf. A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 w*x > 3 y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211922 *)
(* Peter J. C. Moses, Apr 13 2012 *)
Previous Showing 21-30 of 202 results. Next

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

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