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A190020 Number of obtuse triangles on an n X n grid (or geoboard). 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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.

FORMULA

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

CROSSREFS

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

Sequence in context: A125387 A126545 A193168 * A213560 A159506 A081863

Adjacent sequences:  A190017 A190018 A190019 * A190021 A190022 A190023

KEYWORD

nonn

AUTHOR

Martin Renner, May 04 2011

EXTENSIONS

Extended by Ray Chandler, May 04 2011

STATUS

approved

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Last modified September 19 13:31 EDT 2021. Contains 347563 sequences. (Running on oeis4.)