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

1,7

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.

Links

Table of n, a(n) for n=1..85.

Michel Lagneau, Triangles

Eric Weisstein's World of Mathematics, Circumradius

Eric Weisstein's World of Mathematics, Inradius

Example

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}.

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

Crossrefs

Cf. A009003, A188158, A208984, A210207.

Sequence in context: A119788 A334561 A059592 * A319099 A098087 A354987

Adjacent sequences: A295551 A295552 A295553 * A295555 A295556 A295557

Keyword

nonn

Author

Michel Lagneau, Feb 03 2018

Status

approved