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.

Showing 1-5 of 5 results.

A212959 Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.

Original entry on oeis.org

1, 4, 11, 20, 33, 48, 67, 88, 113, 140, 171, 204, 241, 280, 323, 368, 417, 468, 523, 580, 641, 704, 771, 840, 913, 988, 1067, 1148, 1233, 1320, 1411, 1504, 1601, 1700, 1803, 1908, 2017, 2128, 2243, 2360, 2481, 2604, 2731, 2860, 2993, 3128, 3267
Offset: 0

Views

Author

Clark Kimberling, Jun 01 2012

Keywords

Comments

In the following guide to related sequences: M=max(x,y,z), m=min(x,y,z), and R=range=M-m. In some cases, it is an offset of the listed sequence which fits the conditions shown for w,x,y. Each sequence satisfies a linear recurrence relation, some of which are identified in the list by the following code (signature):
A: 2, 0, -2, 1, i.e., a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4);
B: 3, -2, -2, 3, -1;
C: 4, -6, 4, -1;
D: 1, 2, -2, -1, 1;
E: 2, 1, -4, 1, 2, -1;
F: 2, -1, 1, -2, 1;
G: 2, -1, 0, 1, -2, 1;
H: 2, -1, 2, -4, 2, -1, 2, -1;
I: 3, -3, 2, -3, 3, -1;
J: 4, -7, 8, -7, 4, -1.
...
A212959 ... |w-x|=|x-y| ...... recurrence type A
A212960 ... |w-x| != |x-y| ................... B
A212683 ... |w-x| < |x-y| .................... B
A212684 ... |w-x| >= |x-y| ................... B
A212963 ... see entry for definition ......... B
A212964 ... |w-x| < |x-y| < |y-w| ............ B
A006331 ... |w-x| < y ........................ C
A005900 ... |w-x| <= y ....................... C
A212965 ... w = R ............................ D
A212966 ... 2*w = R
A212967 ... w < R ............................ E
A212968 ... w >= R ........................... E
A077043 ... w = x > R ........................ A
A212969 ... w != x and x > R ................. E
A212970 ... w != x and x < R ................. E
A055998 ... w = x + y - 1
A011934 ... w < floor((x+y)/2) ............... B
A182260 ... w > floor((x+y)/2) ............... B
A055232 ... w <= floor((x+y)/2) .............. B
A011934 ... w >= floor((x+y)/2) .............. B
A212971 ... w < floor((x+y)/3) ............... B
A212972 ... w >= floor((x+y)/3) .............. B
A212973 ... w <= floor((x+y)/3) .............. B
A212974 ... w > floor((x+y)/3) ............... B
A212975 ... R is even ........................ E
A212976 ... R is odd ......................... E
A212978 ... R = 2*n - w - x
A212979 ... R = average{w,x,y}
A212980 ... w < x + y and x < y .............. B
A212981 ... w <= x+y and x < y ............... B
A212982 ... w < x + y and x <= y ............. B
A212983 ... w <= x + y and x <= y ............ B
A002623 ... w >= x + y and x <= y ............ B
A087811 ... w = 2*x + y ...................... A
A008805 ... w = 2*x + 2*y .................... D
A000982 ... 2*w = x + y ...................... F
A001318 ... 2*w = 2*x + y .................... F
A001840 ... w = 3*x + y
A212984 ... 3*w = x + y
A212985 ... 3*w = 3*x + y
A001399 ... w = 2*x + 3*y
A212986 ... 2*w = 3*x + y
A008810 ... 3*x = 2*x + y .................... F
A212987 ... 3*w = 2*x + 2*y
A001972 ... w = 4*x + y ...................... G
A212988 ... 4*w = x + y ...................... G
A212989 ... 4*w = 4*x + y
A008812 ... 5*w = 2*x + 3*y
A016061 ... n < w + x + y <= 2*n ............. C
A000292 ... w + x + y <=n .................... C
A000292 ... 2*n < w + x + y <= 3*n ........... C
A212977 ... n/2 < w + x + y <= n
A143785 ... w < R < x ........................ E
A005996 ... w < R <= x ....................... E
A128624 ... w <= R <= x ...................... E
A213041 ... R = 2*|w - x| .................... A
A213045 ... R < 2*|w - x| .................... B
A087035 ... R >= 2*|w - x| ................... B
A213388 ... R <= 2*|w - x| ................... B
A171218 ... M < 2*m .......................... B
A213389 ... R < 2|w - x| ..................... E
A213390 ... M >= 2*m ......................... E
A213391 ... 2*M < 3*m ........................ H
A213392 ... 2*M >= 3*m ....................... H
A213393 ... 2*M > 3*m ........................ H
A213391 ... 2*M <= 3*m ....................... H
A047838 ... w = |x + y - w| .................. A
A213396 ... 2*w < |x + y - w| ................ I
A213397 ... 2*w >= |x + y - w| ............... I
A213400 ... w < R < 2*w
A069894 ... min(|w-x|,|x-y|) = 1
A000384 ... max(|w-x|,|x-y|) = |w-y|
A213395 ... max(|w-x|,|x-y|) = w
A213398 ... min(|w-x|,|x-y|) = x ............. A
A213399 ... max(|w-x|,|x-y|) = x ............. D
A213479 ... max(|w-x|,|x-y|) = w+x+y ......... D
A213480 ... max(|w-x|,|x-y|) != w+x+y ........ E
A006918 ... |w-x| + |x-y| > w+x+y ............ E
A213481 ... |w-x| + |x-y| <= w+x+y ........... E
A213482 ... |w-x| + |x-y| < w+x+y ............ E
A213483 ... |w-x| + |x-y| >= w+x+y ........... E
A213484 ... |w-x|+|x-y|+|y-w| = w+x+y
A213485 ... |w-x|+|x-y|+|y-w| != w+x+y ....... J
A213486 ... |w-x|+|x-y|+|y-w| > w+x+y ........ J
A213487 ... |w-x|+|x-y|+|y-w| >= w+x+y ....... J
A213488 ... |w-x|+|x-y|+|y-w| < w+x+y ........ J
A213489 ... |w-x|+|x-y|+|y-w| <= w+x+y ....... J
A213490 ... w,x,y,|w-x|,|x-y| distinct
A213491 ... w,x,y,|w-x|,|x-y| not distinct
A213493 ... w,x,y,|w-x|,|x-y|,|w-y| distinct
A213495 ... w = min(|w-x|,|x-y|,|w-y|)
A213492 ... w != min(|w-x|,|x-y|,|w-y|)
A213496 ... x != max(|w-x|,|x-y|)
A213498 ... w != max(|w-x|,|x-y|,|w-y|)
A213497 ... w = min(|w-x|,|x-y|)
A213499 ... w != min(|w-x|,|x-y|)
A213501 ... w != max(|w-x|,|x-y|)
A213502 ... x != min(|w-x|,|x-y|)
...
A211795 includes a guide for sequences that count 4-tuples (w,x,y,z) having all terms in {0,...,n} and satisfying selected properties. Some of the sequences indexed at A211795 satisfy recurrences that are represented in the above list.
Partial sums of the numbers congruent to {1,3} mod 6 (see A047241). - Philippe Deléham, Mar 16 2014

Examples

			a(1)=4 counts these (x,y,z): (0,0,0), (1,1,1), (0,1,0), (1,0,1).
Numbers congruent to {1, 3} mod 6: 1, 3, 7, 9, 13, 15, 19, ...
a(0) = 1;
a(1) = 1 + 3 = 4;
a(2) = 1 + 3 + 7 = 11;
a(3) = 1 + 3 + 7 + 9 = 20;
a(4) = 1 + 3 + 7 + 9 + 13 = 33;
a(5) = 1 + 3 + 7 + 9 + 13 + 15 = 48; etc. - _Philippe Deléham_, Mar 16 2014

References

  • A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
  • P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

Links

Crossrefs

Cf. A047241, A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] == Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 50]]   (* A212959 *)
  • PARI
    a(n)=(6*n^2+8*n+3)\/4 \\ Charles R Greathouse IV, Jul 28 2015

Formula

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1+2*x+3*x^2)/((1+x)*(1-x)^3).
a(n) + A212960(n) = (n+1)^3.
a(n) = (6*n^2 + 8*n + 3 + (-1)^n)/4. - Luce ETIENNE, Apr 05 2014
a(n) = 2*A069905(3*(n+1)+2) - 3*(n+1). - Ayoub Saber Rguez, Aug 31 2021

A213482 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, 3, 14, 41, 87, 161, 265, 409, 594, 831, 1120, 1473, 1889, 2381, 2947, 3601, 4340, 5179, 6114, 7161, 8315, 9593, 10989, 12521, 14182, 15991, 17940, 20049, 22309, 24741, 27335, 30113, 33064, 36211, 39542, 43081, 46815, 50769, 54929
Offset: 0

Views

Author

Clark Kimberling, Jun 13 2012

Keywords

Comments

For a guide to related sequences, see A212959.

Links

Crossrefs

Cf. A212959, A213479, A213481, A213483.

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]]   (* A213482 *)

Formula

a(n) + A213483(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.: (3*x + 8*x^2 + 10*x^3 + 3*x^4 - x^5)/((1 - x)^4*(1 + x)^2).
From Ayoub Saber Rguez, Dec 29 2021: (Start)
a(n) = A213481(n) - A213479(n).
a(n) = (23*n^3 + 39*n^2 + n + 9 - (3*n+9)*((n+1) mod 2))/24. (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)
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