A093072 -id:A093072 - cp's OEIS frontend

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

Martin Renner, May 04 2011

nonn

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

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.

Cf. A045996, A077435, A093072, A280653.

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

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

Extended by Ray Chandler, May 04 2011

More terms from Lars Blomberg, Feb 26 2017

A190020 Number of obtuse triangles on an n X n grid (or geoboard).

Original entry on oeis.org

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

Offset: 1

Martin Renner, May 04 2011

nonn

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

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.

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

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

Extended by Ray Chandler, May 04 2011

A348668 Decimal expansion of the probability that a triangle formed by three points uniformly and independently chosen at random in a rectangle with dimensions 1 X 2 is obtuse.

Original entry on oeis.org

7, 9, 8, 3, 7, 4, 2, 8, 5, 1, 2, 6, 9, 2, 1, 0, 6, 0, 3, 8, 5, 1, 0, 4, 7, 9, 4, 1, 8, 7, 3, 5, 8, 7, 5, 2, 2, 8, 6, 3, 1, 6, 5, 8, 3, 0, 2, 0, 5, 0, 9, 4, 1, 1, 0, 1, 8, 9, 2, 4, 4, 6, 9, 7, 0, 2, 8, 8, 4, 0, 5, 3, 9, 5, 2, 8, 3, 8, 7, 3, 1, 3, 8, 5, 4, 2, 8, 9, 5, 8, 3, 6, 8, 1, 6, 1, 4, 1, 5, 7, 2, 7, 1, 0, 2

Offset: 0

Amiram Eldar, Oct 29 2021

The problem of calculating this probability was proposed by Hawthorne (1955) and solved by Langford (1969, 1970). It was mentioned as an unsolved problem in Ogilvy (1962).

			0.79837428512692106038510479418735875228631658302050...

A. M. Mathai, An introduction to geometrical probability: distributional aspects with applications, Amsterdam: Gordon and Breach, 1999, pp. 250-253.
Paul J. Nahin, Digital Dice: Computational Solutions to Practical Probability Problems, Princeton University Press, 2008, pp. 8-11.
Luis A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley, 1976, pp. 21-22.
C. Stanley Ogilvy, Tomorrow's Math: Unsolved Problems for the Amateur, Oxford University Press, New York, 1962, p. 114.

Frank Hawthorne, Problem E1150, The American Mathematical Monthly, Vol. 62, No. 1 (1955), p. 40; Obtuse triangle within a rectangle, Solution to Problem E1150, ibid., Vol. 78, No. 4 (1971), p. 405.
Eric Langford, The probability that a random triangle is obtuse, Biometrika, Vol. 56, No. 3 (1969), p. 689.
Eric Langford, A problem in geometric probability, Mathematics Magazine, Vol 43, No. 5 (1970), pp. 237-244.

Cf. A093072, A341942.

RealDigits[1199/1200 + 13*Pi/128 - 3*Log[2]/4, 10, 100][[1]]

1199/1200 + 13*Pi/128 - 3*log(2)/4 \\ Michel Marcus, Oct 29 2021

Equals 1199/1200 + 13*Pi/128 - 3*log(2)/4.

cp's OEIS Frontend

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

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A190020 Number of obtuse triangles on an n X n grid (or geoboard).

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A348668 Decimal expansion of the probability that a triangle formed by three points uniformly and independently chosen at random in a rectangle with dimensions 1 X 2 is obtuse.

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