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

Author

Martin Renner, May 04 2011

Keywords

• nonn

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.

Crossrefs

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

Formula

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

Extensions

Extended by Ray Chandler, May 04 2011