A212959 -id:A212959 - cp's OEIS frontend

A212960 Number of (w,x,y) with all terms in {0,...,n} and |w-x| != |x-y|.

0, 4, 16, 44, 92, 168, 276, 424, 616, 860, 1160, 1524, 1956, 2464, 3052, 3728, 4496, 5364, 6336, 7420, 8620, 9944, 11396, 12984, 14712, 16588, 18616, 20804, 23156, 25680, 28380, 31264, 34336, 37604, 41072, 44748, 48636, 52744, 57076

Offset: 0

Clark Kimberling, Jun 01 2012

a(n)+A212959(n)=(n+1)^3. Every term is divisible by 4.

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A212959.

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, 45]]   (* A212960 *)
m/4 (* integers *)

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)=4x(1+x^2+x^3) and g(x)=(1+x)(1-x)^4.
a(n) = (4*n^3 + 6*n^2 + 4*n+1 - (-1)^n)/4. - Luce ETIENNE, Apr 05 2014
E.g.f.: (x*(7 + 9*x + 2*x^2)*cosh(x) + (1 + 7*x + 9*x^2 + 2*x^3)*sinh(x))/2. - Stefano Spezia, Aug 11 2025

A212963 a(n) = number of ordered triples (w,x,y) such that w,x,y are all in {0,...,n} and the numbers |w-x|, |x-y|, |y-w| are distinct.

Original entry on oeis.org

0, 0, 0, 12, 36, 84, 156, 264, 408, 600, 840, 1140, 1500, 1932, 2436, 3024, 3696, 4464, 5328, 6300, 7380, 8580, 9900, 11352, 12936, 14664, 16536, 18564, 20748, 23100, 25620, 28320, 31200, 34272, 37536, 41004, 44676, 48564, 52668, 57000

Offset: 0

Clark Kimberling, Jun 02 2012

For each n, there are (n+1)^3 ordered triples, ranging in lexicographical order from (0,0,0) to (n,n,n).  For n = 3, the ordered triples (w,x,y) for which |w-x|, |x-y|, |y-w| are distinct are listed in the Example.
For a guide to related sequences, see A212959.
The ambiguous term "ordered triple" here means that the order matters: (w,x,y) is a different triple from (w,y,x), etc. It does not mean that wN. J. A. Sloane, Dec 28 2021

			a(3) counts the 12 ordered triples in the first column of the following list:
(w,x,y) (|w-x|,|x-y|,|y-w|)
----------------------------
(0,1,3)      (1,2,3)
(0,2,3)      (2,1,3)
(0,3,1)      (3,2,1)
(0,3,2)      (3,1,2)
(1,0,3)      (1,3,2)
(1,3,0)      (2,3,1)
(2,0,3)      (2,3,1)
(2,3,0)      (1,3,2)
(3,0,1)      (3,1,2)
(3,0,2)      (3,2,1)
(3,1,0)      (2,1,3)
(3,2,0)      (1,2,3)

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A002623, A212959.

t = Compile[{{n, _Integer}},
Module[{s = 0}, (Do[If[Abs[w - x] != Abs[x - y] && Abs[x - y] != Abs[y - w] &&
Abs[y - w] != Abs[w - x], s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (*A212963*)
m/12 (*essentially A002623*)

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
G.f.: 12*x^3/((1 + x)*(1 - x)^4).
a(n+3) = 12*A002623(n).
a(n) = (2*n^3 - 3*n^2 - 2*n + 3*(n mod 2))/2. - Ayoub Saber Rguez, Dec 06 2021

Definition corrected by Clark Kimberling, Dec 28 2021

A212966 Number of (w,x,y) with all terms in {0,...,n} and 2*w=range{w,x,y}.

Original entry on oeis.org

1, 1, 3, 8, 10, 12, 23, 25, 29, 44, 48, 52, 73, 77, 83, 108, 114, 120, 151, 157, 165, 200, 208, 216, 257, 265, 275, 320, 330, 340, 391, 401, 413, 468, 480, 492, 553, 565, 579, 644, 658, 672, 743, 757, 773, 848, 864, 880, 961, 977, 995, 1080, 1098

Offset: 0

Clark Kimberling, Jun 02 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (0,1,2,0,-2,-1,0,1).

Cf. A212959.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 w == Max[w, x, y] - Min[w, x, y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A212966 *)

a(n) = a(n-2)+2*a(n-3)-2*a(n-5)-a(n-6)+a(n-8).

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

A212980 Number of (w,x,y) with all terms in {0,...,n} and w

Original entry on oeis.org

0, 1, 6, 17, 37, 68, 113, 174, 254, 355, 480, 631, 811, 1022, 1267, 1548, 1868, 2229, 2634, 3085, 3585, 4136, 4741, 5402, 6122, 6903, 7748, 8659, 9639, 10690, 11815, 13016, 14296, 15657, 17102, 18633, 20253, 21964, 23769, 25670, 27670

Offset: 1

Clark Kimberling, Jun 03 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A002620, A212981, A212959.

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]]   (* A212980 *)

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
G.f.: (x + 3*x^2 + x^3)/((1 + x)*(1 - x)^4).
From Ayoub Saber Rguez, Oct 08 2021: (Start)
a(n) = A212981(n) - A002620(n+1).
a(n) = (10*n^3 + 15*n^2 + 2*n - 3 + 3*((n+1) mod 2))/24. (End)

A212984 Number of (w,x,y) with all terms in {0..n} and 3w = x+y.

Original entry on oeis.org

1, 1, 3, 6, 8, 12, 17, 21, 27, 34, 40, 48, 57, 65, 75, 86, 96, 108, 121, 133, 147, 162, 176, 192, 209, 225, 243, 262, 280, 300, 321, 341, 363, 386, 408, 432, 457, 481, 507, 534, 560, 588, 617, 645, 675, 706, 736, 768, 801, 833, 867, 902, 936, 972, 1009

Offset: 0

Clark Kimberling, Jun 04 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A212959.

[1 + Floor(2*n/3) + Floor(n^2/3) : n in [0..80]]; // Wesley Ivan Hurt, Jul 25 2016

A212984:=n->1 + floor(2*n/3) + floor(n^2/3): seq(A212984(n), n=0..100); # Wesley Ivan Hurt, Jul 25 2016

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[3 w == x + y, s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 70]]   (* A212984 *)

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4.
G.f.: f(x)/g(x), where f(x) = 1 - x + 2*x^2 and g(x) = (1+x+x^2)*(1-x)^3.
a(n) = 1 + floor(2*n/3) + floor(n^2/3). - Wesley Ivan Hurt, Jul 25 2016

A212985 Number of (w,x,y) with all terms in {0,...,n} and 3w = 3x + y.

Original entry on oeis.org

1, 2, 3, 7, 9, 11, 18, 21, 24, 34, 38, 42, 55, 60, 65, 81, 87, 93, 112, 119, 126, 148, 156, 164, 189, 198, 207, 235, 245, 255, 286, 297, 308, 342, 354, 366, 403, 416, 429, 469, 483, 497, 540, 555, 570, 616, 632, 648, 697, 714, 731, 783, 801, 819, 874

Offset: 0

Clark Kimberling, Jun 04 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A212959.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[3 w == 3 x + y, s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 70]]   (* A212985 *)

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7).

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

A212987 Number of (w,x,y) with all terms in {0,...,n} and 3w = 2x+2*y.

Original entry on oeis.org

1, 1, 3, 5, 8, 10, 16, 18, 25, 29, 37, 41, 52, 56, 68, 74, 87, 93, 109, 115, 132, 140, 158, 166, 187, 195, 217, 227, 250, 260, 286, 296, 323, 335, 363, 375, 406, 418, 450, 464, 497, 511, 547, 561, 598, 614, 652, 668, 709, 725, 767, 785, 828, 846, 892

Offset: 0

Clark Kimberling, Jun 04 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (0,2,1,-1,-2,0,1).

Cf. A212959.

t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[3 w == 2 x + 2 y, s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 70]]   (* A212987 *)

a(n) = 2*a(n-2)+a(n-3)-a(n-4)-2*a(n-5)+a(n-7).

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

A212988 Number of (w,x,y) with all terms in {0,...,n} and 4*w = x+y.

Original entry on oeis.org

1, 1, 2, 4, 7, 9, 12, 16, 21, 25, 30, 36, 43, 49, 56, 64, 73, 81, 90, 100, 111, 121, 132, 144, 157, 169, 182, 196, 211, 225, 240, 256, 273, 289, 306, 324, 343, 361, 380, 400, 421, 441, 462, 484, 507, 529, 552, 576, 601, 625, 650, 676, 703, 729, 756, 784

Offset: 0

Clark Kimberling, Jun 04 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A212959.

t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[4 w == x + y, s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 70]]   (* A212988 *)

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

A212989 Number of (w,x,y) with all terms in {0,...,n} and 4w = 4x+y.

Original entry on oeis.org

1, 2, 3, 4, 9, 11, 13, 15, 24, 27, 30, 33, 46, 50, 54, 58, 75, 80, 85, 90, 111, 117, 123, 129, 154, 161, 168, 175, 204, 212, 220, 228, 261, 270, 279, 288, 325, 335, 345, 355, 396, 407, 418, 429, 474, 486, 498, 510, 559, 572, 585, 598, 651, 665, 679, 693

Offset: 0

Clark Kimberling, Jun 04 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 2, -2, 0, 0, -1, 1).

Cf. A212959.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[4 w == 4 x + y, s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 70]]   (* A212989 *)
LinearRecurrence[{1,0,0,2,-2,0,0,-1,1},{1,2,3,4,9,11,13,15,24},60] (* Harvey P. Dale, Sep 20 2023 *)

a(n) = a(n-1)+2*a(n-3)-2*a(n-4)-a(n-7)+a(n-8).

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

A213041 Number of triples (w,x,y) with all terms in {0..n} and 2*|w-x| = max(w,x,y) - min(w,x,y).

Original entry on oeis.org

1, 2, 7, 12, 21, 30, 43, 56, 73, 90, 111, 132, 157, 182, 211, 240, 273, 306, 343, 380, 421, 462, 507, 552, 601, 650, 703, 756, 813, 870, 931, 992, 1057, 1122, 1191, 1260, 1333, 1406, 1483, 1560, 1641, 1722, 1807, 1892, 1981, 2070, 2163, 2256

Offset: 0

Clark Kimberling, Jun 10 2012

See A212959 for a guide to related sequences.

For n > 3, a(n-2) is the number of distinct values of the magic constant in a perimeter-magic (n-1)-gon of order n (see A342819). - Stefano Spezia, Mar 23 2021

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 10-13).
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A002620, A004526, A058331, A212959, A168277 (first differences), A342819.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Max[w, x, y] - Min[w, x, y] == 2 Abs[w - x],
s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (* A213041 *)

Vec((1+3*x^2)/((1-x)^3*(1+x)) + O(x^99)) \\ Altug Alkan, May 06 2016

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3.
G.f.: (1 + 3*x^2)/((1 - x)^3 * (1 + x)).
a(n) = (n+1)^2 - 2*A004526(n-1) - 2. - Wesley Ivan Hurt, Jul 15 2013
a(n) = A002620(n+2) + 3*A002620(n). - R. J. Mathar, Jul 15 2013
a(n)+a(n+1) = A058331(n+1). - R. J. Mathar, Jul 15 2013
a(n) = n*(n+1) + (1+(-1)^n)/2. - Wesley Ivan Hurt, May 06 2016
E.g.f.: x*(x + 2)*exp(x) + cosh(x). - Ilya Gutkovskiy, May 06 2016
a(n) = A000384(n+1) - A137932(n+2). - Federico Provvedi, Aug 17 2023

cp's OEIS Frontend

A212960 Number of (w,x,y) with all terms in {0,...,n} and |w-x| != |x-y|.

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A212963 a(n) = number of ordered triples (w,x,y) such that w,x,y are all in {0,...,n} and the numbers |w-x|, |x-y|, |y-w| are distinct.

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A212966 Number of (w,x,y) with all terms in {0,...,n} and 2*w=range{w,x,y}.

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A212980 Number of (w,x,y) with all terms in {0,...,n} and w

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A212984 Number of (w,x,y) with all terms in {0..n} and 3w = x+y.

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A212985 Number of (w,x,y) with all terms in {0,...,n} and 3w = 3x + y.

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A212987 Number of (w,x,y) with all terms in {0,...,n} and 3*w = 2*x+2*y.

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A212988 Number of (w,x,y) with all terms in {0,...,n} and 4*w = x+y.

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A212987 Number of (w,x,y) with all terms in {0,...,n} and 3w = 2x+2*y.

A212989 Number of (w,x,y) with all terms in {0,...,n} and 4w = 4x+y.