A280652 -id:A280652 - cp's OEIS frontend

A279413 Triangle read by rows: T(n,k), n>=k>=1, is the number of isosceles triangles with integer coordinates that have a bounding box of size n X k.

0, 0, 4, 0, 2, 12, 0, 0, 6, 16, 0, 2, 4, 6, 24, 0, 0, 2, 8, 10, 28, 0, 2, 4, 2, 8, 6, 36, 0, 0, 2, 0, 6, 8, 10, 40, 0, 2, 4, 2, 12, 10, 8, 10, 56, 0, 0, 2, 4, 2, 4, 10, 8, 10, 60, 0, 2, 4, 2, 4, 2, 12, 6, 12, 6, 60, 0, 0, 2, 0, 2, 4, 6, 12, 6, 8, 14, 64, 0, 2

Offset: 1

Lars Blomberg, Feb 16 2017

			Triangle begins:
0
0, 4
0, 2, 12
0, 0, 6, 16
0, 2, 4, 6, 24
0, 0, 2, 8, 10, 28
0, 2, 4, 2, 8, 6, 36
0, 0, 2, 0, 6, 8, 10, 40
0, 2, 4, 2, 12, 10, 8, 10, 56
0, 0, 2, 4, 2, 4, 10, 8, 10, 60
0, 2, 4, 2, 4, 2, 12, 6, 12, 6, 60
0, 0, 2, 0, 2, 4, 6, 12, 6, 8, 14, 64
0, 2, 4, 2, 4, 6, 8, 10, 16, 14, 12, 14, 72
0, 0, 2, 0, 2, 4, 2, 8, 14, 4, 6, 12, 18, 76
0, 2, 4, 2, 4, 2, 8, 2, 8, 10, 16, 10, 12, 10, 84
0, 0, 2, 0, 6, 4, 2, 4, 6, 16, 6, 4, 10, 12, 14, 88
0, 2, 4, 2, 4, 2, 8, 2, 16, 6, 16, 10, 16, 6, 24, 10, 104
0, 0, 2, 0, 2, 0, 2, 4, 6, 4, 10, 12, 10, 12, 10, 12, 14, 100
0, 2, 4, 2, 4, 2, 12, 6, 4, 6, 12, 10, 20, 6, 12, 14, 16, 10, 124
0, 0, 2, 0, 2, 0, 2, 0, 2, 4, 6, 12, 10, 12, 10, 12, 18, 12, 10, 112
-----
Denote by 'o' the point adjacent to the two equal sides, and by 'x' the other two.
n=4, k=3:
...x  x...  .o..  ..o.  x...  ...x
o...  ...o  ...x  x...  ...x  x...
...x  x...  x...  ...x  .o..  ..o.
So T(4,3)=6.
-----
n=4,k=4:
o...  ...o  .x..  ..x.  o...  ...o  ..x.  .x..
...x  x...  ....  ....  ....  ....  ...x  x...
....  ....  ...x  x...  ...x  x...  ....  ....
.x..  ..x.  o...  ...o  ..x.  .x..  o...  ...o
-
...x  x...  x...  ...x  o..x  x..o  x...  ...x
.o..  ..o.  ....  ....  ....  ....  ....  ....
....  ....  .o..  ..o.  ....  ....  ....  ....
x...  ...x  ...x  x...  x...  ...x  o..x  x..o
So T(4,4)=16.

Lars Blomberg, Table of n, a(n) for n = 1..9870 (the first 140 rows)
Lars Blomberg, Algorithms for computing A279413, A279414, A186434 and A271908

Cf. A186434, A187452, A271910-A271913, A271915, A279414.
See A279415 for right isosceles triangles.
See A280639 for obtuse isosceles triangles.
See A279418 for acute isosceles triangles.
See A279433 for all right triangles.
See A280652 for all obtuse triangles.
See A280653 for all acute triangles.
See A279432 for all triangles.

A279418 Triangle read by rows: T(n,k), n>=k>=1, is the number of acute isosceles triangles with integer coordinates that have a bounding box of size n X k.

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 0, 0, 2, 8, 0, 0, 2, 2, 16, 0, 0, 2, 4, 6, 16, 0, 0, 2, 0, 4, 2, 24, 0, 0, 2, 0, 2, 4, 6, 24, 0, 0, 2, 0, 6, 6, 4, 6, 40, 0, 0, 2, 0, 2, 0, 6, 4, 6, 40, 0, 0, 2, 0, 2, 0, 8, 2, 8, 2, 40, 0, 0, 2, 0, 2, 4, 2, 8, 2, 4, 10, 40, 0, 0, 2, 0, 2, 0

Offset: 1

Lars Blomberg, Feb 27 2017

			Triangle begins:
0
0,0
0,0,8
0,0,2,8
0,0,2,2,16
0,0,2,4,6,16
0,0,2,0,4,2,24
0,0,2,0,2,4,6,24
0,0,2,0,6,6,4,6,40
0,0,2,0,2,0,6,4,6,40
0,0,2,0,2,0,8,2,8,2,40
0,0,2,0,2,4,2,8,2,4,10,40
0,0,2,0,2,0,2,2,8,10,8,10,48
0,0,2,0,2,4,2,4,10,0,2,8,14,48
0,0,2,0,2,0,6,0,4,6,12,6,8,6,56
0,0,2,0,2,0,2,0,2,8,2,0,6,8,10,56
------
The vertex between the two equal sides is 'o'.
For n=3, k=3:
x.x   x..   o..   .x.   .x.   .o.   ..o   ..x
...   ..o   ..x   x..   ..x   ...   x..   o..
.o.   x..   .x.   ..o   o..   x.x   .x.   ..x
So T(3,3)=8
------
For n=6, k=4:
x....o   o....x   .x....   ....x.
......   ......   ......   ......
......   ......   ......   ......
.x....   ....x.   x....o   o....x
So T(6,4)=4

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

Cf. A190317.
See A279415 for right isosceles triangles.
See A280639 for obtuse isosceles triangles.
See A279413 for all isosceles triangles.
See A279433 for all right triangles.
See A280652 for all obtuse triangles.
See A280653 for all acute triangles.
See A279432 for all triangles.

A279433 Triangle read by rows: T(n,k), n>=k>=1, is the number of right triangles with integral coordinates that have a bounding box of size n X k.

Original entry on oeis.org

0, 0, 4, 0, 6, 4, 0, 4, 12, 4, 0, 4, 6, 12, 12, 0, 4, 8, 12, 12, 4, 0, 4, 4, 6, 12, 20, 4, 0, 4, 4, 12, 12, 12, 20, 4, 0, 4, 4, 4, 14, 12, 20, 12, 12, 0, 4, 4, 4, 12, 12, 16, 12, 12, 20, 0, 4, 4, 8, 8, 6, 12, 20, 20, 20, 4, 0, 4, 4, 4, 4, 12, 28, 12, 12, 12

Offset: 1

Lars Blomberg, Feb 27 2017

			Triangle begins:
0
0,4
0,6,4
0,4,12,4
0,4,6,12,12
0,4,8,12,12,4
0,4,4,6,12,20,4
0,4,4,12,12,12,20,4
0,4,4,4,14,12,20,12,12
0,4,4,4,12,12,16,12,12,20
0,4,4,8,8,6,12,20,20,20,4
0,4,4,4,4,12,28,12,12,12,20,4
0,4,4,4,4,12,6,20,20,16,20,20,12
0,4,4,4,12,4,24,12,12,12,20,12,20,4
0,4,4,4,4,4,12,6,28,20,12,20,20,20,4
0,4,4,4,4,4,8,12,20,20,12,20,12,20,28,4
0,4,4,4,4,12,4,12,18,12,20,12,28,12,20,20,28
-----
The right angle is 'o'.
For n=2, k=2:
ox   xo   x.   .x
x.   .x   ox   xo
So T(2,2)=4
-----
For n=3, k=2:
o.x   x.x   x.o   x..   .o.   ..x
x..   .o.   ..x   o.x   x.x   x.o
So T(3,2)=6

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

Cf. A077435.
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 A280652 for all obtuse triangles.
See A280653 for all acute triangles.
See A279432 for all triangles.

A280639 Triangle read by rows: T(n,k), n>=k>=1, is the number of obtuse isosceles triangles with integral coordinates that have a bounding box of size n X k.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 8, 0, 2, 2, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 12, 0, 2, 2, 2, 4, 0, 0, 0, 12, 0, 0, 0, 4, 0, 0, 0, 0, 0, 16, 0, 2, 2, 2, 2, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 0, 2, 2, 2, 2, 6, 4

Offset: 1

Lars Blomberg, Feb 27 2017

			Triangle begins:
0
0,0
0,0,0
0,0,0,4
0,2,0,0,4
0,0,0,0,0,8
0,2,2,0,0,0,8
0,0,0,0,0,0,0,12
0,2,2,2,4,0,0,0,12
0,0,0,4,0,0,0,0,0,16
0,2,2,2,2,0,0,0,0,0,16
0,0,0,0,0,0,0,0,0,0,0,20
0,2,2,2,2,6,4,4,4,0,0,0,20
-------
The obtuse angle is 'o'.
For n=4, k=4:
x...   x...   ...x   ...x
..o.   ....   .o..   ....
....   .o..   ....   ..o.
...x   ...x   x...   x...
So T(4,4)=4
-------
For n=5, k=2:
x...x   ..o..
..o..   x...x
So T(5,2)=2

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

Cf. A190318.
See A279415 for right 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 A280653 for all acute triangles.
See A279432 for all triangles.

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.

Original entry on oeis.org

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.

A279415 Triangle read by rows: T(n,k), n>=k>=1, is the number of right isosceles triangles with integral coordinates that have a bounding box of size n X k.

Original entry on oeis.org

0, 0, 4, 0, 2, 4, 0, 0, 4, 4, 0, 0, 2, 4, 4, 0, 0, 0, 4, 4, 4, 0, 0, 0, 2, 4, 4, 4, 0, 0, 0, 0, 4, 4, 4, 4, 0, 0, 0, 0, 2, 4, 4, 4, 4, 0, 0, 0, 0, 0, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 2, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 2, 4, 4

Offset: 1

Lars Blomberg, Feb 27 2017

			Triangle begins:
0
0,4
0,2,4
0,0,4,4
0,0,2,4,4
0,0,0,4,4,4
0,0,0,2,4,4,4
0,0,0,0,4,4,4,4
0,0,0,0,2,4,4,4,4
0,0,0,0,0,4,4,4,4,4
0,0,0,0,0,2,4,4,4,4,4
0,0,0,0,0,0,4,4,4,4,4,4
0,0,0,0,0,0,2,4,4,4,4,4,4
-------
The right angle is 'o'.
For n=4, k=3:
x...   .o..   ..o.   ...x
...x   ...x   x...   x...
.o..   x...   ...x   ..o.
So T(4,3)=4
-------
For n=4, k=4:
o..x   x..o   x...   ...x
....   ....   ....   ....
....   ....   ....   ....
x...   ...x   o..x   x..o
So T(4,4)=4

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

Cf. A187452.
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 A280653 for all acute triangles.
See A279432 for all triangles.

A279432 Triangle read by rows: T(n,k), n>=k>=1, is the number of triangles with integer coordinates that have a bounding box of size n X k.

Original entry on oeis.org

0, 0, 4, 0, 10, 20, 0, 16, 34, 48, 0, 22, 44, 70, 88, 0, 28, 58, 88, 118, 140, 0, 34, 68, 102, 140, 178, 204, 0, 40, 82, 124, 166, 208, 250, 280, 0, 46, 92, 142, 184, 238, 284, 334, 368, 0, 52, 106, 156, 214, 268, 318, 376, 430, 468, 0, 58, 116, 178, 236, 290

Offset: 1

Lars Blomberg, Feb 27 2017

T(n,k) = A279433(n,k) + A280652(n,k) + A280653(n,k).

It appears that the main diagonal is 4*A000326.

			Triangle begins:
0
0,4
0,10,20
0,16,34,48
0,22,44,70,88
0,28,58,88,118,140
0,34,68,102,140,178,204
0,40,82,124,166,208,250,280
0,46,92,142,184,238,284,334,368
0,52,106,156,214,268,318,376,430,468
0,58,116,178,236,290,356,418,476,538,580
0,64,130,196,262,328,394,460,526,592,658,704
-----
A right angle is marked 'r', an obtuse one 'o'.
For n=2, k=2
rx   xr   x.   .x
x.   .x   rx   xr
So T(2,2)=4.
-----
For n=3, k=2
xo.   r.x   x.x   x.r   x..   x..   .ox   .r.   ..x   ..x
..x   x..   .r.   ..x   r.x   .ox   x..   x.x   xo.   x.r
So T(3,2)=10.

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

Cf. A045996.
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 A280653 for all acute triangles.

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

cp's OEIS Frontend

A279413 Triangle read by rows: T(n,k), n>=k>=1, is the number of isosceles triangles with integer coordinates that have a bounding box of size n X k.

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A279418 Triangle read by rows: T(n,k), n>=k>=1, is the number of acute isosceles triangles with integer coordinates that have a bounding box of size n X k.

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A279433 Triangle read by rows: T(n,k), n>=k>=1, is the number of right triangles with integral coordinates that have a bounding box of size n X k.

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A280639 Triangle read by rows: T(n,k), n>=k>=1, is the number of obtuse isosceles triangles with integral coordinates that have a bounding box of size n X k.

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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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A279415 Triangle read by rows: T(n,k), n>=k>=1, is the number of right isosceles triangles with integral coordinates that have a bounding box of size n X k.

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A279432 Triangle read by rows: T(n,k), n>=k>=1, is the number of triangles with integer coordinates that have a bounding box of size n X k.

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

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