A190019 -id:A190019 - cp's OEIS frontend

A280653 Triangle read by rows: T(n,k), n>=k>=1, is the number of acute triangles with integer coordinates that have a bounding box of size n X k.

0, 0, 0, 0, 0, 8, 0, 0, 6, 24, 0, 0, 6, 22, 40, 0, 0, 2, 20, 46, 64, 0, 0, 2, 20, 44, 70, 96, 0, 0, 2, 8, 42, 76, 98, 136, 0, 0, 2, 8, 34, 74, 104, 138, 176, 0, 0, 2, 8, 22, 72, 110, 148, 186, 208, 0, 0, 2, 4, 18, 56, 112, 146, 188, 234, 264, 0, 0, 2, 4, 18

Offset: 1

Lars Blomberg, Feb 25 2017

It appears that the main diagonal is 8*A014811.

			Triangle begins:
0
0,0
0,0,8
0,0,6,24
0,0,6,22,40
0,0,2,20,46,64
0,0,2,20,44,70,96
0,0,2,8,42,76,98,136
0,0,2,8,34,74,104,138,176
0,0,2,8,22,72,110,148,186,208
0,0,2,4,18,56,112,146,188,234,264
0,0,2,4,18,44,94,152,198,244,286,328
0,0,2,4,18,32,86,150,196,254,296,342,392
-----
For n=3, k=3:
o.o   o..   o..   .o.   .o.   .o.   ..o   ..o
...   ..o   ..o   o..   ..o   ...   o..   o..
.o.   o..   .o.   ..o   o..   o.o   .o.   ..o
so T(3,3)=8
-----
For n=4, k=3:
o..o   o..o   o...   .o..   ..o.   ...o
....   ....   ...o   ....   ....   o...
.o..   ..o.   o...   o..o   o..o   ...o
so T(4,3)=6
-----
For n=6, k=3:
o.....   .....o
.....o   o.....
o.....   .....o
so T(6,3)=2

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

Cf. A190019.
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 A280652 for all obtuse triangles.
See A279432 for all triangles.

A103429 (1/4)*number of acute 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

2, 194, 3434, 29356, 162190, 679654, 2323878, 6839595, 17909922, 42675551, 94125356, 194693240, 381214450, 712191373, 1277323894, 2210486280, 3706015236, 6040816887, 9601083812, 14916225896, 22701123860, 33905935285

Offset: 1

Hugo Pfoertner, Feb 08 2005

nonn

Cf. all triangles in lattice cube A103426; special triangles in lattice cube: A103427, A103428, A103499, A103500, A103501; A103158 tetrahedra in lattice cube.

Cf. A190019 (analogous 2-dimensional problem).

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

A241224 Number of acute triangles on a centered hexagonal grid of size n.

Original entry on oeis.org

0, 8, 204, 1788, 8690, 30360, 85194, 205394, 441876, 870912, 1601708, 2783574, 4616220, 7358312, 11339430, 16972182, 24763604, 35328426, 49405944, 67873484, 91762128, 122276784

Offset: 1

Martin Renner, Apr 17 2014

nonn

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

			For n = 2 the eight acute triangles are the following:
/. *     * *     * .     . .     . .     . .     * .     . *
. * *   . * .   * * .   * * .   . * .   . * *   . . *   * . .
\. .     . .     . .     * .     * *     . *     * .     . *

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

Cf. A190019, A241223.

a(n) = A241223(n) - A241225(n) - A241226(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

cp's OEIS Frontend

A280653 Triangle read by rows: T(n,k), n>=k>=1, is the number of acute triangles with integer coordinates that have a bounding box of size n X k.

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A103429 (1/4)*number of acute triangles that can be formed from the points of an (n+1)X(n+1)X(n+1) lattice cube.

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

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A241224 Number of acute 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).