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

1,3

Links

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

Eric Weisstein's World of Mathematics, Geoboard.

Eric Weisstein's World of Mathematics, Acute Triangle.

Formula

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

Example

For n = 3 the two acute 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:
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:

Crossrefs

Cf. A028419, A189979, A190022.

Sequence in context: A178129 A203298 A161463 * A014285 A331756 A330152

Adjacent sequences: A190018 A190019 A190020 * A190022 A190023 A190024

Keyword

nonn

Author

Martin Renner, May 04 2011

Extensions

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

Status

approved