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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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 A257099

Adjacent sequences:  A295551 A295552 A295553 * A295555 A295556 A295557

KEYWORD

nonn

AUTHOR

Michel Lagneau, Feb 03 2018

STATUS

approved

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Last modified July 29 14:43 EDT 2021. Contains 346346 sequences. (Running on oeis4.)