A190022 Number of obtuse triangles, distinct up to congruence, on an n X n grid (or geoboard).

0, 0, 2, 12, 39, 95, 193, 355, 597, 943, 1426, 2071, 2904, 3977, 5306, 6956, 8963, 11370, 14225, 17587, 21515, 26053, 31310, 37282, 44061, 51785, 60436, 70127, 80939, 92952, 106267, 120982, 137124, 154841, 174225, 195366, 218394, 243457, 270505, 299749, 331441

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, Obtuse Triangle.

Formula

a(n) = A028419(n) - A190021(n) - A189979(n).

Example

For n = 3 the two obtuse 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:
IsObtuseTriangle:=proc(T) if T[1]^2+T[2]^2<T[3]^2 or T[1]^2+T[3]^2<T[2]^2 or T[2]^2+T[3]^2<T[1]^2 then true else false fi: end:
a:=proc(n) local TriangleSet, ObtuseTriangleSet, i; TriangleSet:=Triangles(n): ObtuseTriangleSet:={}: for i from 1 to nops(TriangleSet) do if IsObtuseTriangle(TriangleSet[i]) then ObtuseTriangleSet:={op(ObtuseTriangleSet), TriangleSet[i]} fi: od: return(nops(ObtuseTriangleSet)); end:

Crossrefs

Cf. A028419, A189979, A190021.

Cf. A103428.

Sequence in context: A026575 A048349 A009632 * A086602 A019006 A363123

Adjacent sequences: A190019 A190020 A190021 * A190023 A190024 A190025

Keyword

nonn

Author

Martin Renner, May 04 2011

Extensions

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

Status

approved