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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

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

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Last modified December 7 03:44 EST 2022. Contains 358649 sequences. (Running on oeis4.)