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

Author

Martin Renner, May 03 2011

Keywords

• nonn
• more

Examples

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

Links

• Alec Jones, Examples for n = 1 to 5
• Alec Jones, Example for n = 24

Crossrefs

Cf. A028419, A186434.

Programs

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:

Extensions

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