A271908 -id:A271908 - 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

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.

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).

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