A190019 Number of acute triangles on an n X n grid (or geoboard).

0, 0, 8, 80, 404, 1392, 3880, 9208, 19536, 38096, 69288, 119224, 196036, 310008, 474336, 705328, 1023216, 1451904, 2020232, 2762848, 3719420, 4937200, 6469424, 8378184, 10734664, 13618168, 17119288, 21338760, 26390452, 32400592, 39508656, 47870200, 57655752

Offset

1,3

Comments

Place all bounding boxes of A280653 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) * A280653(i,j) where k=1 when i=j and k=2 otherwise. - Lars Blomberg, Feb 26 2017

According to Langford (p. 243), the leading order is (53/150-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..5000

Margherita Barile, Geoboard.

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

Eric Weisstein's World of Mathematics, Acute Triangle.

Formula

a(n) = A045996(n) - A077435(n) - A190020(n).

Crossrefs

Cf. A045996, A077435, A093072, A280653.

Cf. A103429 (analogous problem on a 3-dimensional grid).

Sequence in context: A100431 A173116 A102698 * A342353 A055346 A159710

Adjacent sequences: A190016 A190017 A190018 * A190020 A190021 A190022

Keyword

nonn

Author

Martin Renner, May 04 2011

Extensions

Extended by Ray Chandler, May 04 2011

More terms from Lars Blomberg, Feb 26 2017

Status

approved