A189978 Number of isosceles triangles, distinct up to congruence, on an n X n grid (or geoboard).

0, 1, 5, 11, 22, 35, 53, 70, 100, 126, 159, 188, 237, 276, 328, 372, 439, 491, 564, 623, 706, 775, 859, 931, 1049, 1129, 1231, 1323, 1448, 1540, 1674, 1772, 1928, 2041, 2183, 2301, 2483, 2602, 2758, 2898, 3095

Offset

1,3

Links

Table of n, a(n) for n=1..41.

Alec Jones, Examples for n = 1 to 5

Alec Jones, Example for n = 24

Example

For n=3 the five isosceles triangles are:
**.  *.*  .*.  ..*  *..
*..  ...  *..  *..  ..*
...  *..  .*.  ..*  .*.

Maple

Triangles:=proc(n) local TriangleSet, i, j, k, l, A, B, C; TriangleSet:={}: for i from 0 to n do for j from 0 to n do for k from 0 to n do for l from 0 to n do A:=i^2+j^2: B:=k^2+l^2: C:=(i-k)^2+(j-l)^2: if A^2+B^2+C^2<>2*(A*B+B*C+C*A) then TriangleSet:={op(TriangleSet), sort([sqrt(A), sqrt(B), sqrt(C)])}: fi: od: od: od: od: return(TriangleSet); end:
IsIsoscelesTriangle:=proc(T) if T[1]=T[2] or T[1]=T[3] or T[2]=T[3] then true else false fi: end:
a:=proc(n) local TriangleSet, IsoscelesTriangleSet, i;
TriangleSet:=Triangles(n): IsoscelesTriangleSet:={}: for i from 1 to nops(TriangleSet) do if IsIsoscelesTriangle(TriangleSet[i]) then IsoscelesTriangleSet:={op(IsoscelesTriangleSet), TriangleSet[i]} fi: od: return(nops(IsoscelesTriangleSet)); end:

Crossrefs

Cf. A028419, A186434.

Sequence in context: A069097 A222548 A024921 * A192761 A152533 A228485

Adjacent sequences: A189975 A189976 A189977 * A189979 A189980 A189981

Keyword

nonn,more

Author

Martin Renner, May 03 2011

Extensions

a(21)-a(40) from Martin Renner, May 08 2011

Status

approved