A190021 Number of acute triangles, distinct up to congruence, on an n X n grid (or geoboard).

0, 0, 2, 8, 23, 51, 101, 179, 295, 460, 688, 988, 1382, 1876, 2495, 3258, 4191, 5298, 6613, 8166, 9973, 12065, 14472, 17208, 20341, 23873, 27838, 32282, 37238, 42734, 48840, 55573, 62973, 71067, 79934, 89640, 100172, 111613, 123959, 137336, 151842

Offset: 1

Author

Martin Renner, May 04 2011

Keywords

• nonn

Examples

			For n = 3 the two acute triangles are:
*..   .*.
..*   *..
*..   ..*
		

Links

• Eric Weisstein's World of Mathematics, Geoboard.
• Eric Weisstein's World of Mathematics, Acute Triangle.

Crossrefs

Cf. A028419, A189979, A190022.

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:
IsAcuteTriangle:=proc(T) if T[1]^2+T[2]^2>T[3]^2 and T[1]^2+T[3]^2>T[2]^2 and T[2]^2+T[3]^2>T[1]^2 then true else false fi: end:
a:=proc(n) local TriangleSet,AcuteTriangleSet,i; TriangleSet:=Triangles(n): AcuteTriangleSet:={}: for i from 1 to nops(TriangleSet) do if IsAcuteTriangle(TriangleSet[i]) then AcuteTriangleSet:={op(AcuteTriangleSet),TriangleSet[i]} fi: od: return(nops(AcuteTriangleSet)); end:

Formula

a(n) = A028419(n) - A189979(n) - A190022(n).

Extensions

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