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Number of non-congruent Heronian triangles with base length n whose apex lies on or within a space bounded by a semicircle of diameter n.
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%I #10 Feb 28 2016 08:01:47

%S 0,0,0,0,1,0,0,1,0,1,0,0,1,0,2,1,2,0,0,3,1,0,0,2,3,2,0,0,2,5,0,1,0,2,

%T 2,0,4,0,4,5,2,2,0,2,3,0,0,4,0,4,6,6,3,0,1,2,0,3,0,8,2,0,2,1,8,1,0,4,

%U 1,4,0,2,5,6,9,0,1,5,0,8,0,4,0,2,9

%N Number of non-congruent Heronian triangles with base length n whose apex lies on or within a space bounded by a semicircle of diameter n.

%e a(20)=3 as there are 3 non-congruent Heronian triangles with base length of 20 whose apex lies on or within the space bounded by a semicircle of diameter 20. The integer triples are (7,15,20), (11,13,20), (12,16,20).

%t gettriangles[c_] := (s=(a+b+c)/2; area2=s(s-a)(s-b)(s-c); n=0; Do[If[IntegerQ[Sqrt[area2]]&&a+b>c&&a^2+b^2<=c^2, n++], {b, 1, c}, {a, 1, b}]; n); Table[gettriangles[m], {m, 1, 200}]

%Y Cf. A236384.

%K nonn

%O 1,15

%A _Frank M Jackson_, Jan 28 2014