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

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, 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, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 5, 12, 1, 1, 5, 1, 1

Offset: 1

Author

Michel Lagneau, Feb 03 2018

Keywords

• nonn

Comments

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?

a(m) > 1 for m = 7, 18, 26, 31, 35, ... and {A009003(m)} = {25, 50, 65, 75, 85, ...} = {A009177}.

We observe geometric properties:

If a(n) = 1, the unique triangle is a right triangle.

If a(n) = 5, we find two right triangles, two isosceles triangles and another triangle (neither isosceles nor right triangle).

If a(n) = 10, we find three right triangles, two isosceles triangles and five other triangles.

If a(n) = 12, we find four right triangles and eight other triangles.

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.

The inradius r is given by r = A/s and the circumradius is given by R = u*v*w/4A.

Examples

			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}.
		

Links

• Felix Huber, Illustration of the term a(7)
• Michel Lagneau, Triangles
• Eric Weisstein's World of Mathematics, Circumradius
• Eric Weisstein's World of Mathematics, Inradius

Crossrefs

Cf. A009003, A188158, A208984, A210207.

Programs

Mathematica

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