A187452 -id:A187452 - cp's OEIS frontend

A186434 Number of isosceles triangles that can be formed from the n^2 points of n X n grid of points (or geoboard).

0, 4, 36, 148, 444, 1064, 2200, 4024, 6976, 11284, 17396, 25620, 36812, 51216, 69672, 92656, 121392, 156092, 198364, 248292, 307988, 377816, 459072, 552216, 660704, 784076, 924340, 1082228, 1261132, 1460408, 1684464, 1931800, 2208368

Offset: 1

Martin Renner, Apr 10 2011, Apr 13 2011

nonn

This counts triples of distinct points A,B,C such that A,B,C are the vertices of an isosceles triangle with nonzero area. It would be nice to have a formula. - N. J. A. Sloane, Apr 22 2016

Place all bounding boxes of A279413 that will fit into the n X n grid in all possible positions, and the proper rectangles in two orientations: a(n) = Sum_{i=1..n} Sum_{j=1..i} k * (n-i+1) * (n-j+1) * A279413(i,j) where k=1 when i=j and k=2 otherwise. - Lars Blomberg, Feb 20 2017

Lars Blomberg, Table of n, a(n) for n = 1..10000 (the first 67 terms from Nathaniel Johnston)
Margherita Barile, MathWorld -- Geoboard.
Nathaniel Johnston, C program for computing terms

Cf. A045996, A077435, A187452, A189416, A271910-A271913, A271915, A279413, A279414.

Dividing by 4 gives A271908.

with(linalg):
IsTriangle:=proc(points) local a,b,c; a:=points[3]-points[2]: b:=points[3]-points[1]: c:=points[2]-points[1]: if evalf(norm(a,2)+norm(b,2))>evalf(norm(c,2)) and evalf(norm(a,2)+norm(c,2))>evalf(norm(b,2)) and evalf(norm(b,2)+norm(c,2))>evalf(norm(a,2)) then true: else false: fi: end:
IsIsoscelesTriangle:=proc(points) local a,b,c; a:=points[3]-points[2]: b:=points[3]-points[1]: c:=points[2]-points[1]: if IsTriangle(points) then if norm(a,2)=norm(b,2) or norm(a,2)=norm(c,2) or norm(b,2)=norm(c,2) then true: else false: fi: else false: fi; end:
a:=proc(n) local P,TriangleSet,i,j,a,b,c; P:=[]: for i from 0 to n do for j from 0 to n do P:=[op(P),[i,j]]: od; od; TriangleSet:={}: for a from 1 to nops(P) do for b from a+1 to nops(P) do for c from b+1 to nops(P) do if IsIsoscelesTriangle([P[a],P[b],P[c]]) then TriangleSet:={op(TriangleSet),[P[a],P[b],P[c]]}; fi; od; od; od; return(nops(TriangleSet)): end:

a(10)-a(33) from Nathaniel Johnston, Apr 25 2011

A271910 Array read by antidiagonals: T(n,k) = number of ways to choose 3 distinct points from an n X k rectangular grid so that they form an isosceles triangle.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 0, 10, 10, 0, 0, 16, 36, 16, 0, 0, 24, 68, 68, 24, 0, 0, 32, 108, 148, 108, 32, 0, 0, 42, 150, 248, 248, 150, 42, 0, 0, 52, 200, 360, 444, 360, 200, 52, 0, 0, 64, 252, 488, 672, 672, 488, 252, 64, 0, 0, 76, 312, 620, 932, 1064, 932, 620, 312, 76, 0

Offset: 1

N. J. A. Sloane, Apr 24 2016

The triangle must have nonzero area (three collinear points don't count).

			Initial rows of the array:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 4, 10, 16, 24, 32, 42, 52, 64, 76, ...
0, 10, 36, 68, 108, 150, 200, 252, 312, 374, ...
0, 16, 68, 148, 248, 360, 488, 620, 768, 924, ...
0, 24, 108, 248, 444, 672, 932, 1204, 1512, 1836, ...
0, 32, 150, 360, 672, 1064, 1510, 1984, 2524, 3092, ...
0, 42, 200, 488, 932, 1510, 2200, 2944, 3792, 4690, ...
0, 52, 252, 620, 1204, 1984, 2944, 4024, 5256, 6568, ...
0, 64, 312, 768, 1512, 2524, 3792, 5256, 6976, 8816, ...
0, 76, 374, 924, 1836, 3092, 4690, 6568, 8816, 11284, ...
...
As a triangle:
0,
0, 0,
0, 4, 0,
0, 10, 10, 0,
0, 16, 36, 16, 0,
0, 24, 68, 68, 24, 0,
0, 32, 108, 148, 108, 32, 0,
0, 42, 150, 248, 248, 150, 42, 0,
0, 52, 200, 360, 444, 360, 200, 52, 0,
0, 64, 252, 488, 672, 672, 488, 252, 64, 0,
...
To illustrate T(2,3)=10: Label the points
1 2 3
4 5 6
There are 8 small isosceles triangles like 124 plus 135 and 246, for a total of 10.

Chai Wah Wu, Table of n, a(n) for n = 1..3003
Chai Wah Wu, Counting the number of isosceles triangles in rectangular regular grids, arXiv:1605.00180 [math.CO], 2016.

Rows 2,3,4,5 are A271911, A271912, A271913, A271915.
Main diagonal = A186434.
Cf. A187452, A271914.

It appears that for each n >= 2, there is a number K(n) such that row n satisfies the recurrence a(k) = 2*a(k-1)-2*a(k-3)+a(k-4) for k >= K(n). This is based on the fact that the conjectured generating functions for rows 2, 3, 4, 5 have the same denominator, and on Colin Barker's conjectured recurrence for A271911. K(n) is determined by the degree of the numerator of the g.f.

Above conjecture about the recurrence is true for K(n) = (n-1)^2+4 if n is even and K(n) = (n-1)^2+3 if n is odd and not true for smaller K(n). See paper in links. - Chai Wah Wu, May 07 2016

A077435 Number of right triangles whose vertices are lattice points in {1,2,...,n} X {1,2,...,n}.

Original entry on oeis.org

0, 4, 44, 200, 596, 1444, 2960, 5520, 9496, 15332, 23596, 34936, 50020, 69732, 94816, 126176, 164960, 212372, 269620, 337960, 418716, 513444, 623736, 751152, 897776, 1065220, 1255460, 1470680, 1713052, 1984564, 2288304, 2626160, 3000960, 3415124, 3871108

Offset: 1

John W. Layman, Nov 30 2002

nonn

It would be nice to have a formula. - N. J. A. Sloane, Jun 29 2016

			For n=2 if the four points are labeled
  ab
  cd
then the right triangles are abc, abd, acd, bcd, so a(2)=4.
For n=3, label the points
  abc
  def
  ghi
The right triangles are: abd (4*4 ways), acg (4 ways), acd and adf (8 ways each), ace and dbf (4 ways each), for a total of a(3) = 44. - _N. J. A. Sloane_, Jun 30 2016

Lars Blomberg, Table of n, a(n) for n = 1..10000 (the first 184 terms from R. H. Hardin)

Cf. A187452, A279433.

Place all bounding boxes of A279433 that will fit into the n X n grid in all possible positions, and the proper rectangles in two orientations: a(n) = Sum_{i=1..n} Sum_{j=1..i} k * (n-i+1) * (n-j+1) * A279433(i,j) where k=1 when i=j and k=2 otherwise. - Lars Blomberg, Mar 01 2017

a(1) corrected by Lars Blomberg, Mar 01 2017

A279413 Triangle read by rows: T(n,k), n>=k>=1, is the number of isosceles triangles with integer coordinates that have a bounding box of size n X k.

Original entry on oeis.org

0, 0, 4, 0, 2, 12, 0, 0, 6, 16, 0, 2, 4, 6, 24, 0, 0, 2, 8, 10, 28, 0, 2, 4, 2, 8, 6, 36, 0, 0, 2, 0, 6, 8, 10, 40, 0, 2, 4, 2, 12, 10, 8, 10, 56, 0, 0, 2, 4, 2, 4, 10, 8, 10, 60, 0, 2, 4, 2, 4, 2, 12, 6, 12, 6, 60, 0, 0, 2, 0, 2, 4, 6, 12, 6, 8, 14, 64, 0, 2

Offset: 1

Lars Blomberg, Feb 16 2017

			Triangle begins:
0
0, 4
0, 2, 12
0, 0, 6, 16
0, 2, 4, 6, 24
0, 0, 2, 8, 10, 28
0, 2, 4, 2, 8, 6, 36
0, 0, 2, 0, 6, 8, 10, 40
0, 2, 4, 2, 12, 10, 8, 10, 56
0, 0, 2, 4, 2, 4, 10, 8, 10, 60
0, 2, 4, 2, 4, 2, 12, 6, 12, 6, 60
0, 0, 2, 0, 2, 4, 6, 12, 6, 8, 14, 64
0, 2, 4, 2, 4, 6, 8, 10, 16, 14, 12, 14, 72
0, 0, 2, 0, 2, 4, 2, 8, 14, 4, 6, 12, 18, 76
0, 2, 4, 2, 4, 2, 8, 2, 8, 10, 16, 10, 12, 10, 84
0, 0, 2, 0, 6, 4, 2, 4, 6, 16, 6, 4, 10, 12, 14, 88
0, 2, 4, 2, 4, 2, 8, 2, 16, 6, 16, 10, 16, 6, 24, 10, 104
0, 0, 2, 0, 2, 0, 2, 4, 6, 4, 10, 12, 10, 12, 10, 12, 14, 100
0, 2, 4, 2, 4, 2, 12, 6, 4, 6, 12, 10, 20, 6, 12, 14, 16, 10, 124
0, 0, 2, 0, 2, 0, 2, 0, 2, 4, 6, 12, 10, 12, 10, 12, 18, 12, 10, 112
-----
Denote by 'o' the point adjacent to the two equal sides, and by 'x' the other two.
n=4, k=3:
...x  x...  .o..  ..o.  x...  ...x
o...  ...o  ...x  x...  ...x  x...
...x  x...  x...  ...x  .o..  ..o.
So T(4,3)=6.
-----
n=4,k=4:
o...  ...o  .x..  ..x.  o...  ...o  ..x.  .x..
...x  x...  ....  ....  ....  ....  ...x  x...
....  ....  ...x  x...  ...x  x...  ....  ....
.x..  ..x.  o...  ...o  ..x.  .x..  o...  ...o
-
...x  x...  x...  ...x  o..x  x..o  x...  ...x
.o..  ..o.  ....  ....  ....  ....  ....  ....
....  ....  .o..  ..o.  ....  ....  ....  ....
x...  ...x  ...x  x...  x...  ...x  o..x  x..o
So T(4,4)=16.

Lars Blomberg, Table of n, a(n) for n = 1..9870 (the first 140 rows)
Lars Blomberg, Algorithms for computing A279413, A279414, A186434 and A271908

Cf. A186434, A187452, A271910-A271913, A271915, A279414.
See A279415 for right isosceles triangles.
See A280639 for obtuse isosceles triangles.
See A279418 for acute isosceles triangles.
See A279433 for all right triangles.
See A280652 for all obtuse triangles.
See A280653 for all acute triangles.
See A279432 for all triangles.

A108279 a(n) = number of squares with corners on an n X n grid, distinct up to congruence.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 15, 18, 23, 28, 33, 38, 45, 51, 58, 65, 73, 80, 89, 97, 107, 116, 126, 134, 146, 158, 169, 180, 192, 204, 218, 228, 243, 257, 270, 285, 302, 316, 331, 346, 364, 379, 397, 414, 433, 451, 468, 484, 505, 523, 544, 563, 584, 603, 625

Offset: 1

Hugo Pfoertner, Jun 05 2005

nonn

Number of different sizes occurring among the A002415(n) = n^2*(n^2-1)/12 squares that can be drawn using points of an n X n square array as corners.

a(n) is also the number of rectangular isosceles triangles, distinct up to congruence, on an n X n grid (or geoboard). - Martin Renner, May 03 2011

			a(3)=3 because the 6 different squares that can be drawn on a 3 X 3 square lattice come in 3 sizes:
  4 squares of side length 1:
  x.x.o    o.x.x    o.o.o    o.o.o
  x.x.o    o.x.x    x.x.o    o.x.x
  o.o.o    o.o.o    x.x.o    o.x.x
  1 square of side length sqrt(2):
  o.x.o
  x.o.x
  o.x.o
  1 square of side length 2:
  x.o.x
  o.o.o
  x.o.x
.
a(4)=5 because there are 5 different sizes of squares that can be drawn using the points of a 4 X 4 square lattice:
  x.x.o.o    o.x.o.o    x.o.x.o    o.x.o.o    x.o.o.x
  x.x.o.o    x.o.x.o    o.o.o.o    o.o.o.x    o.o.o.o
  o.o.o.o    o.x.o.o    x.o.x.o    x.o.o.o    o.o.o.o
  o.o.o.o    o.o.o.o    o.o.o.o    o.o.x.o    x.o.o.x

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Henry Bottomley, Illustration of initial terms of A002415

Cf. A002415, A024206, A187452.

a[n_] := Module[{v = Table[0, (n - 1)^2]}, Do[v[[k^2 + (w - k)^2]] = 1, {w, 1, n - 1}, {k, 0, w - 1}]; Total[v]]; Array[a, 55](* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

a(n) = my(v=vector((n-1)^2)); for(w=1, n-1, for(k=0, w-1, v[k^2+(w-k)^2]=1)); vecsum(v); \\ Andrew Howroyd, Sep 17 2017

More terms from David W. Wilson, Jun 07 2005

A279415 Triangle read by rows: T(n,k), n>=k>=1, is the number of right isosceles triangles with integral coordinates that have a bounding box of size n X k.

Original entry on oeis.org

0, 0, 4, 0, 2, 4, 0, 0, 4, 4, 0, 0, 2, 4, 4, 0, 0, 0, 4, 4, 4, 0, 0, 0, 2, 4, 4, 4, 0, 0, 0, 0, 4, 4, 4, 4, 0, 0, 0, 0, 2, 4, 4, 4, 4, 0, 0, 0, 0, 0, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 2, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 2, 4, 4

Offset: 1

Lars Blomberg, Feb 27 2017

			Triangle begins:
0
0,4
0,2,4
0,0,4,4
0,0,2,4,4
0,0,0,4,4,4
0,0,0,2,4,4,4
0,0,0,0,4,4,4,4
0,0,0,0,2,4,4,4,4
0,0,0,0,0,4,4,4,4,4
0,0,0,0,0,2,4,4,4,4,4
0,0,0,0,0,0,4,4,4,4,4,4
0,0,0,0,0,0,2,4,4,4,4,4,4
-------
The right angle is 'o'.
For n=4, k=3:
x...   .o..   ..o.   ...x
...x   ...x   x...   x...
.o..   x...   ...x   ..o.
So T(4,3)=4
-------
For n=4, k=4:
o..x   x..o   x...   ...x
....   ....   ....   ....
....   ....   ....   ....
x...   ...x   o..x   x..o
So T(4,4)=4

Lars Blomberg, Table of n, a(n) for n = 1..9870 (the first 140 rows)

Cf. A187452.
See A280639 for obtuse isosceles triangles.
See A279418 for acute isosceles triangles.
See A279413 for all isosceles triangles.
See A279433 for all right triangles.
See A280652 for all obtuse triangles.
See A280653 for all acute triangles.
See A279432 for all triangles.

A334581 Number of ways to choose 3 points that form an equilateral triangle from the A000292(n) points in a regular tetrahedral grid of side length n.

Original entry on oeis.org

0, 0, 4, 24, 84, 224, 516, 1068, 2016, 3528, 5832, 9256, 14208, 21180, 30728, 43488, 60192, 81660, 108828, 142764, 184708, 236088, 298476, 373652, 463524, 570228, 696012, 843312, 1014720, 1213096, 1441512, 1703352, 2002196, 2341848, 2726400, 3160272, 3648180

Offset: 0

Peter Kagey, May 06 2020

nonn

a(n) >= 4 * A269747(n).
a(n) >= 4 * A000389(n+3) = A210569(n+2).
a(n) >= 4 * (n-1) + 4 * a(n-1) - 6 * a(n-2) + 4 * a(n-3) - a(n-4) for n >= 4.

Peter Kagey, Table of n, a(n) for n = 0..1000 (Computed via Anders Kaseorg's program in the Stack Exchange link.)
Code Golf Stack Exchange, Triangles in a tetrahedron

Cf. A000292, A000389, A210569.
Cf. A000332 (equilateral triangles in triangular grid), A269747 (regular tetrahedra in a tetrahedral grid), A102698 (equilateral triangles in cube), A103158 (regular tetrahedra in cube).
Cf. A077435, A085582, A187452, A189412-A189418, A271910.

A190317 Number of acute isosceles triangles on an n X n grid.

Original entry on oeis.org

0, 0, 8, 48, 164, 448, 976, 1864, 3328, 5512, 8640, 12984, 18836, 26576, 36584, 49056, 64560, 83640, 106904, 134816, 168004, 206952, 252480, 305360, 366312, 436120, 515864, 605928, 707644, 822120, 950216, 1092784, 1251784, 1427848, 1622416, 1835968, 2070404

Offset: 1

Martin Renner, May 08 2011

nonn

Place all bounding boxes of A279418 that will fit into the n X n grid in all possible positions, and the proper rectangles in two orientations: a(n) = Sum(i=1..n, Sum(j=1..i, k * (n-i+1) * (n-j+1) * A279418(i,j))) where k=1 when i=j and k=2 otherwise. - Lars Blomberg, Mar 02 2017

Lars Blomberg, Table of n, a(n) for n = 1..10000 (the first 100 terms from Chai Wah Wu)

Cf. A186434, A187452, A190318, A279418.

a(n) = A186434(n) - A187452(n) - A190318(n).

a(10)-a(37) from Nathaniel Johnston, May 09 2011

A190318 Number of obtuse isosceles triangles on an n X n grid.

Original entry on oeis.org

0, 0, 0, 4, 36, 100, 256, 496, 968, 1672, 2736, 4092, 6188, 8764, 12144, 16464, 22224, 28928, 37400, 47076, 59244, 73580, 90344, 109000, 132048, 158000, 187528, 220716, 259348, 301388, 350088, 402792, 463176, 529720, 602888, 683092, 774476, 872100, 978232

Offset: 1

Martin Renner, May 08 2011

nonn

Place all bounding boxes of A280639 that will fit into the n X n grid in all possible positions, and the proper rectangles in two orientations: a(n) = Sum(i=1..n, Sum(j=1..i, k * (n-i+1) * (n-j+1) * A280639(i,j))) where k=1 when i=j and k=2 otherwise. - Lars Blomberg, Mar 02 2017

Lars Blomberg, Table of n, a(n) for n = 1..10000 (the first 100 terms from Chai Wah Wu)
Nathaniel Johnston, C program for computing terms

Cf. A186434, A187452, A190317, A280639.

a(n) = A186434(n) - A190317(n) - A187452(n).

a(10)-a(39) from Nathaniel Johnston, May 09 2011

A271913 Number of ways to choose three distinct points from a 4 X n grid so that they form an isosceles triangle.

Original entry on oeis.org

0, 16, 68, 148, 248, 360, 488, 620, 768, 924, 1096, 1272, 1464, 1660, 1872, 2088, 2320, 2556, 2808, 3064, 3336, 3612, 3904, 4200, 4512, 4828, 5160, 5496, 5848, 6204, 6576, 6952, 7344, 7740, 8152, 8568, 9000, 9436, 9888, 10344, 10816, 11292, 11784, 12280, 12792, 13308, 13840, 14376, 14928, 15484

Offset: 1

N. J. A. Sloane, Apr 24 2016

nonn

Chai Wah Wu, Counting the number of isosceles triangles in rectangular regular grids, arXiv:1605.00180 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Row 4 of A271910.

Cf. A186434, A187452.

Join[{0, 16, 68, 148, 248, 360, 488, 620}, LinearRecurrence[{2, 0, -2, 1}, {768, 924, 1096, 1272}, 42]] (* Jean-François Alcover, Sep 03 2018 *)

Conjectured g.f.: 4*x*(x^10-x^8+2*x^6+x^5+4*x^4+4*x^3-3*x^2-9*x-4)/((x+1)*(x-1)^3).
Conjectured recurrence: a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n > 12.
Conjectures from Colin Barker, Apr 25 2016: (Start)
a(n) = -3/2*(143+(-1)^n)+64*n+5*n^2 for n>8.
a(n) = 5*n^2+64*n-216 for n>8 and even.
a(n) = 5*n^2+64*n-213 for n>8 and odd.
(End)
The conjectured g.f. and recurrence are true. See paper in links. - Chai Wah Wu, May 07 2016

More terms from Jean-François Alcover, Sep 03 2018

cp's OEIS Frontend

A186434 Number of isosceles triangles that can be formed from the n^2 points of n X n grid of points (or geoboard).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Extensions

A271910 Array read by antidiagonals: T(n,k) = number of ways to choose 3 distinct points from an n X k rectangular grid so that they form an isosceles triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A077435 Number of right triangles whose vertices are lattice points in {1,2,...,n} X {1,2,...,n}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A279413 Triangle read by rows: T(n,k), n>=k>=1, is the number of isosceles triangles with integer coordinates that have a bounding box of size n X k.

Views

Author

Keywords

Examples

Links

Crossrefs

A108279 a(n) = number of squares with corners on an n X n grid, distinct up to congruence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A279415 Triangle read by rows: T(n,k), n>=k>=1, is the number of right isosceles triangles with integral coordinates that have a bounding box of size n X k.

Views

Author

Keywords

Examples

Links

Crossrefs

A334581 Number of ways to choose 3 points that form an equilateral triangle from the A000292(n) points in a regular tetrahedral grid of side length n.

Views

Author

Keywords

Comments

Links

Crossrefs

A190317 Number of acute isosceles triangles on an n X n grid.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A190318 Number of obtuse isosceles triangles on an n X n grid.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula