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making obtuse triangles

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Working with lengths that are whole units from 1 - 100, how many obtuse triangles can be formed?

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I get fewer ...

Spoiler

47308.

To avoid double counting, I restricted a <= b <= c.

I let a be in [1, 100] and b in [a, 100]. I then counted the integers c in [ sqrt {(a2)+(b2)},  a+b -1 ].
I also noticed 52 right triangles (Pythagorean triples) .

This is a plot of possible obtuse triangles for each value of a between 1 and 100.

When a = 27 the number reaches a maximum of 1166.

obs100.JPG

 

 

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Case by case results, inviting algorithm falsification, while limiting lengths to 10.

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Rows: values of a; Columns: values of b; Elements: triangle counts.
      OBTUSE 10
0 0 0 0 0 0 0 0 0 0
0 1 1 1 1 1 1 1 1 0
0 0 1 1 2 2 2 2 1 0
0 0 0 2 2 2 2 2 1 0
0 0 0 0 2 3 2 1 0 0
0 0 0 0 0 2 1 0 0 0
0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0


Row sums (obtuse triangles for 10 values of a)
0 8 11 11 8 3 1 0 0 0  <= showing no possible obtuse triangles when 2> a >7

Total triangles
42

More matrices, for maximum lengths of 2-9:

Spoiler

      OBTUSE 2
0 0
0 0

      OBTUSE 3
0 0 0
0 1 0
0 0 0

      OBTUSE 4
0 0 0 0
1 1 0   <= meaning a=b=2 permits (only) one obtuse triangle: c=3
0 0 0 0
0 0 0 0

      OBTUSE 5
0 0 0 0 0
0 1 1 1 0  <= meaning a=2 b=4 permits (only) one obtuse triangle: c=5
0 0 1 0 0  <= showing that a=3 b=4 c=5 (right-)triangle was not counted
0 0 0 0 0
0 0 0 0 0

      OBTUSE 6
0 0 0 0 0 0
1 1 1 1 0  <= counting (only) 223, 234, 245, 256 for a=2
0 0 1 1 1 0  <= indicating a=3 b=4 c=6 obtuse triangle (only) was counted; not c=5,7
0 0 0 1 0 0
0 0 0 0 0 0
0 0 0 0 0 0

      OBTUSE 7
0 0 0 0 0 0 0
0 1 1 1 1 1 0
0 0 1 1 2 1 0
0 0 0 2 1 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0

      OBTUSE 8
0 0 0 0 0 0 0 0
0 1 1 1 1 1 1 0
0 0 1 1 2 2 1 0
0 0 0 2 2 1 0 0
0 0 0 0 1 1 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0

      OBTUSE 9
0 0 0 0 0 0 0 0 0
0 1 1 1 1 1 1 1 0
0 0 1 1 2 2 2 1 0
0 0 0 2 2 2 1 1 0
0 0 0 0 2 2 1 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0

 

 

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