A190318 -id:A190318 - cp's OEIS frontend

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.

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.

A190317 Number of acute isosceles triangles on an n X n grid.

Original entry on oeis.org

0, 0, 8, 48, 164, 448, 976, 1864, 3328, 5512, 8640, 12984, 18836, 26576, 36584, 49056, 64560, 83640, 106904, 134816, 168004, 206952, 252480, 305360, 366312, 436120, 515864, 605928, 707644, 822120, 950216, 1092784, 1251784, 1427848, 1622416, 1835968, 2070404

Offset: 1

Martin Renner, May 08 2011

nonn

Place all bounding boxes of A279418 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) * A279418(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)

Cf. A186434, A187452, A190318, A279418.

a(n) = A186434(n) - A187452(n) - A190318(n).

a(10)-a(37) from Nathaniel Johnston, May 09 2011

A241230 Number of obtuse isosceles triangles on a centered hexagonal grid of size n.

Original entry on oeis.org

0, 12, 108, 528, 1674, 4104, 8622, 16206, 28332, 46200, 71484, 106746, 153552, 214860, 293166, 391470, 512592, 660546, 840840, 1054704, 1308696, 1606776

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

Cf. A190318.

a(n) = A241228(n) - A241229(n).

a(7) from Martin Renner, May 31 2014

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

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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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A190317 Number of acute isosceles triangles on an n X n grid.

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A241230 Number of obtuse isosceles triangles on a centered hexagonal grid of size n.

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