A190020 -id:A190020 - cp's OEIS frontend

A280652 Triangle read by rows: T(n,k), n>=k>=1, is the number of obtuse triangles with integer coordinates that have a bounding box of size n X k.

0, 0, 0, 0, 4, 8, 0, 12, 16, 20, 0, 18, 32, 36, 36, 0, 24, 48, 56, 60, 72, 0, 30, 62, 76, 84, 88, 104, 0, 36, 76, 104, 112, 120, 132, 140, 0, 42, 86, 130, 136, 152, 160, 184, 180, 0, 48, 100, 144, 180, 184, 192, 216, 232, 240, 0, 54, 110, 166, 210, 228, 232

Offset: 1

Lars Blomberg, Feb 26 2017

			Triangle begins:
0
0,0
0,4,8
0,12,16,20
0,18,32,36,36
0,24,48,56,60,72
0,30,62,76,84,88,104
0,36,76,104,112,120,132,140
0,42,86,130,136,152,160,184,180
0,48,100,144,180,184,192,216,232,240
0,54,110,166,210,228,232,252,268,284,312
0,60,124,188,240,272,272,296,316,336,352,372
0,66,134,202,258,314,328,332,352,372,400,428,436
0,72,148,224,288,352,380,400,408,432,448,480,508,536
-----
The obtuse angle is 'o'.
For n=3, k=2:
xo.   x..   .ox   ..x
..x   .ox   x..   xo.
So T(3,2)=4
-----
For n=3, k=3:
xo.   x..   x..   x..   .ox   ..x   ..x   ..x
...   o..   ..o   ...   ...   o..   ..o   ...
..x   ..x   ..x   .ox   x..   x..   x..   xo.
So T(3,3)=8

Lars Blomberg, Table of n, a(n) for n = 1..9870 (the first 140 rows)

Cf. A190020.
See A279415 for right isosceles triangles.
See A280639 for obtuse isosceles triangles.
See A279418 for acute isosceles triangles.
See A279413 for all isosceles triangles.
See A279433 for all right triangles.
See A280653 for all acute triangles.
See A279432 for all triangles.

A103428 (1/12)*Number of non-degenerate obtuse triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

Original entry on oeis.org

0, 62, 1270, 11266, 63322, 266748, 915720, 2701073, 7077080, 16876415, 37242500, 77038188, 150862354, 281877711, 505585682, 874900010, 1466826558, 2390947859, 3799984292, 5903574820, 8984255594, 13418520513, 19700297034, 28470461533

Offset: 1

Hugo Pfoertner, Feb 08 2005

nonn

Cf. all triangles in lattice cube A103426; special triangles in lattice cube: A103427, A103429, A103499, A103500, A103501; A103158 tetrahedra in lattice cube.

Cf. A190020 (analogous 2-dimensional problem).

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

Original entry on oeis.org

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

A241226 Number of obtuse triangles on a centered hexagonal grid of size n.

Original entry on oeis.org

0, 12, 480, 4488, 22278, 78288, 220374, 531594, 1143648, 2254440, 4145244, 7202190, 11940444, 19029300, 29318754, 43872402, 63999528, 91288794, 127642164, 175326336, 237000816, 315772032

Offset: 1

Martin Renner, Apr 17 2014

A centered hexagonal grid of size n is a grid with A003215(n-1) points forming a hexagonal lattice.

Eric Weisstein's World of Mathematics, Hex Number.
Eric Weisstein's World of Mathematics, Obtuse Triangle.

Cf. A190020.

a(n) = A241223(n) - A241224(n) - A241225(n).

a(7) from Martin Renner, May 31 2014

a(8)-a(22) from Giovanni Resta, May 31 2014

A341942 Decimal expansion of (388 + 15Pi)/(212 - 15Pi).

Original entry on oeis.org

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

Offset: 1

Peter Kagey, Feb 24 2021

The constant is equal to lim_{n -> oo} A190020(n)/A190019(n), which is the ratio of the probability of an obtuse triangle to an acute triangle, when all three vertices are drawn uniformly at random in a square.

			2.6390960417866483205308889755825371915092...

Code Golf Stack Exchange, Ratio of obtuse triangles to acute triangles in the square.
Eric Langford, The probability that a random triangle is obtuse, Biometrika, 56(3), 1969, p. 689.
Plot of A190020(n)/A190019(n) vs n using OEIS Plot2.
Index entries for transcendental numbers

Cf. A000796, A190019, A190020.

RealDigits[(388 + 15*Pi)/(212 - 15*Pi), 10, 100][[1]]

(388 + 15*Pi)/(212 - 15*Pi) \\ Michel Marcus, Feb 25 2021

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A280652 Triangle read by rows: T(n,k), n>=k>=1, is the number of obtuse triangles with integer coordinates that have a bounding box of size n X k.

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A103428 (1/12)*Number of non-degenerate obtuse triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

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

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A241226 Number of obtuse triangles on a centered hexagonal grid of size n.

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A341942 Decimal expansion of (388 + 15*Pi)/(212 - 15*Pi).

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A341942 Decimal expansion of (388 + 15Pi)/(212 - 15Pi).