

A190019


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


6



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
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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 * (ni+1) * (nj+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 (53/150Pi/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, NovDec, 1970, 237244.
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 3dimensional 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



