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A295554 a(n) is the number of distinct integer-sided triangles inscribed in a circle of radius A009003(n) whose inradius are integers.

%I A295554 #60 Feb 16 2025 08:33:52
%S A295554 1,1,1,1,1,1,5,1,1,1,1,1,1,1,1,1,1,5,1,1,1,1,1,1,1,12,1,1,1,1,5,1,1,1,
%T A295554 12,1,1,1,1,1,1,5,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,10,12,1,1,1,1,1,12,
%U A295554 1,1,1,5,1,1,1,1,1,1,1,1,5,12,1,1,5,1,1
%N A295554 a(n) is the number of distinct integer-sided triangles inscribed in a circle of radius A009003(n) whose inradius are integers.
%C A295554 For n <= 200, the number of distinct integer-sided triangles inscribed in a circle of radius A009003(n) whose inradius are integers belongs to the set E = {1, 5, 10, 12, 38} where a(168) = 38 (see the table given in reference). Is the set E infinite when n is infinite?
%C A295554 a(m) > 1 for m = 7, 18, 26, 31, 35, ... and {A009003(m)} = {25, 50, 65, 75, 85, ...} = {A009177}.
%C A295554 We observe geometric properties:
%C A295554 If a(n) = 1, the unique triangle is a right triangle.
%C A295554 If a(n) = 5, we find two right triangles, two isosceles triangles and another triangle (neither isosceles nor right triangle).
%C A295554 If a(n) = 10, we find three right triangles, two isosceles triangles and five other triangles.
%C A295554 If a(n) = 12, we find four right triangles and eight other triangles.
%C A295554 The area A of a triangle whose sides have lengths u, v, and w is given by Heron's formula: A = sqrt(s*(s-u)*(s-v)*(s-w)), where s = (u+v+w)/2.
%C A295554 The inradius r is given by r = A/s and the circumradius is given by R = u*v*w/4A.
%H A295554 Felix Huber, <a href="/A295554/a295554_1.pdf">Illustration of the term a(7)</a>
%H A295554 Michel Lagneau, <a href="/A295554/a295554.pdf">Triangles</a>
%H A295554 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Circumradius.html">Circumradius</a>
%H A295554 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Inradius.html">Inradius</a>
%e A295554 a(7) = 5 because there exists 5 distinct triangles of integer circumradius R = A009003(7)= 25 with the corresponding integer inradius {4, 6, 8, 10, 12}.
%t A295554 A009003=Select[Range[200], Length[PowersRepresentations[#^2, 2, 2]] > 1 &];lst= {};Do[R=Part[A009003,n];it=0;Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s (s-a) (s-b) (s-c);If[area2>0&&IntegerQ[Sqrt[area2]]&&R==a*b*c/(4*Sqrt[area2])&&IntegerQ[Sqrt[area2]/s],it=it+1]],{a,2*R},{b,a},{c,b}];AppendTo[lst,it],{n,1,30}];lst
%Y A295554 Cf. A009003, A188158, A208984, A210207.
%K A295554 nonn
%O A295554 1,7
%A A295554 _Michel Lagneau_, Feb 03 2018