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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 (list; graph; refs; listen; history; text; internal format)
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
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. A103429 (analogous problem on a 3-dimensional grid).
Sequence in context: A100431 A173116 A102698 * A342353 A055346 A159710
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

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Last modified March 29 08:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)