A190317 -id:A190317 - cp's OEIS frontend

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.

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.

A190318 Number of obtuse isosceles triangles on an n X n grid.

Original entry on oeis.org

0, 0, 0, 4, 36, 100, 256, 496, 968, 1672, 2736, 4092, 6188, 8764, 12144, 16464, 22224, 28928, 37400, 47076, 59244, 73580, 90344, 109000, 132048, 158000, 187528, 220716, 259348, 301388, 350088, 402792, 463176, 529720, 602888, 683092, 774476, 872100, 978232

Offset: 1

Martin Renner, May 08 2011

nonn

Place all bounding boxes of A280639 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) * A280639(i,j))) where k=1 when i=j and k=2 otherwise. - Lars Blomberg, Mar 02 2017

Lars Blomberg, Table of n, a(n) for n = 1..10000 (the first 100 terms from Chai Wah Wu)
Nathaniel Johnston, C program for computing terms

Cf. A186434, A187452, A190317, A280639.

a(n) = A186434(n) - A190317(n) - A187452(n).

a(10)-a(39) from Nathaniel Johnston, May 09 2011

A241229 Number of acute isosceles triangles on a centered hexagonal grid of size n.

Original entry on oeis.org

0, 8, 144, 768, 2558, 6564, 14334, 27626, 48828, 80772, 126692, 190230, 275700, 388016, 532386, 714906, 941600, 1219410, 1556184, 1959608, 2438868, 3002700

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, Acute Triangle.
Eric Weisstein's World of Mathematics, Isosceles Triangle.

Cf. A190317.

a(n) = A241228(n) - A241230(n).

a(7) from Martin Renner, May 31 2014

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

cp's OEIS Frontend

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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A190318 Number of obtuse isosceles triangles on an n X n grid.

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A241229 Number of acute isosceles triangles on a centered hexagonal grid of size n.

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