A190020 Number of obtuse triangles on an n X n grid (or geoboard).

0, 0, 24, 236, 1148, 3932, 10760, 25392, 53576, 103824, 188104, 322852, 529116, 835028, 1275360, 1893496, 2742208, 3886568, 5402448, 7381316, 9928860, 13168484, 17243896, 22319864, 28579720, 36237928, 45532720, 56732668

Offset

1,3

Comments

Place all bounding boxes of A280652 that will fit into the n X n grid in all possible positions, and the proper rectangles in two orientations: a(n) = Sum_{i=1..n} Sum_{j=1..i} k * (n-i+1) * (n-j+1) * A280652(i,j) where k=1 when i=j and k=2 otherwise. - Lars Blomberg, Mar 02 2017

According to Langford (p. 243), the leading order is (97/150 + Pi/40)*C(n^2,3). See A093072. - Michael R Peake, Jan 15 2021

Links

Lars Blomberg, Table of n, a(n) for n = 1..10000

Margherita Barile, MathWorld -- Geoboard.

Eric Langford, A problem in geometric probability, Mathematics Magazine, Nov-Dec, 1970, 237-244.

Eric Weisstein's World of Mathematics, Obtuse Triangle.

Formula

a(n) = A045996(n) - A077435(n) - A190019(n).

Crossrefs

Cf. A045996, A077435, A093072, A190019, A280652.

Sequence in context: A125387 A126545 A193168 * A213560 A159506 A081863

Adjacent sequences: A190017 A190018 A190019 * A190021 A190022 A190023

Keyword

nonn

Author

Martin Renner, May 04 2011

Extensions

Extended by Ray Chandler, May 04 2011

Status

approved