
COMMENTS

The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(sa)*(sb)*(sc)), where s = (a+b+c)/2.
The sequence a(n) is the union of four subsequences A, B, C and D where:
A is the subsequence with areas 60, 210, 1716, 2730, 4080, 5814, 7980, 10626, ... where n is odd, and the corresponding sides are of the form (4k, 4k^21, 4k^2+1) with areas 2k(4k^21) for k = 2, 3, 6, 7, 8, 9, 11, ... These areas are in the sequence A069072 (areas of primitive Pythagorean triangles whose odd sides differ by 2).
B is the subsequence with areas 6, 30, 84, 330, 546, 2310, 4914, 6090, ... where n is even, and the corresponding sides are of the form (2k+1, 2k(k+1), 2k(k+1)+1) with areas k(k+1)(2k+1) for k = 1, 2, 3, 5, 6, 7, 10, 13, 14, ... These areas are in the sequence A055112 (Areas of Pythagorean triangles (a, b, c) with c = b+1.
C is the subsequence with areas 84, 198, 1380, 4176, ... where n is odd but the areas are not Pythagorean triangles.
D is the subsequence with areas 24, 60, 210, 924, 1248, 480, 1224, 1938, ... where n is even but the areas are not Pythagorean triangles.
The triangles with the same areas are not unique; for example:
(8, 15, 17) and (6, 25, 29) => A = 60; the first is a Pythagorean triangle, the second is not.
(12, 35, 37) and (7, 65, 68) => A = 210; the first is a Pythagorean triangle, the second is not.
The following table gives the first values (n, A, p, a, b, c) where A is the area of the triangles, p is the perimeter and a, b, c are the sides.
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 n  A  p  a  b  c 
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 4  6  12 = 4*3  3  4  5 
 5  60  40 = 5*8  8  15  17 
 6  30  30 = 6*5  5  12  13 
 7  210  84 = 7*12  12  35  37 
 8  24  32 = 8*4  4  13  15 
 9  84  72 = 9*8  8  29  35 
 10  60  60 = 10*6  6  25  29 
 11  198  132 = 11*12  12  55  65 
 12  330  132 = 12*11  11  60  61 
 13  1716  312 = 13*24  24  143  145 
 14  546  182 = 14*13  13  84  85 
 15  2730  420 = 15*28  28  195  197 
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