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

1,2

Comments

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

Links

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

Maple

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:

Crossrefs

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

Dividing by 4 gives A271908.

Sequence in context: A069053 A193282 A192217 * A270989 A272557 A370660

Adjacent sequences: A186431 A186432 A186433 * A186435 A186436 A186437

Keyword

nonn

Author

Martin Renner, Apr 10 2011, Apr 13 2011

Extensions

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

Status

approved