1,2

It is conjectured that the longest possible side c of a triangle with integer sides and inradius n is given by A057721(n)=n^4+3*n^2+1.

For n >= 1, a(n) >= 1 because triangle (a, b, c) = (n^2+2, n^4+2n^2+1, n^4+3n^2+1) has inradius n. - David W. Wilson, Jun 17 2006

David W. Wilson, Table of n, a(n) for n = 1..10000

Thomas Mautsch, Additional terms

The even numbered terms are given by a(2*n)=A007237(n).

a(n) = sum_{k:k|n} A120252(k)

a(1)=1: {3,4,5} is the only triangle with integer sides and inradius 1.

a(2)=5: {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17} are the only triangles with integer sides and inradius 2.

a(4)=A120252(1)+A120252(2)+A120252(4)=1+4+13 because 1, 2 and 4 are the factors of 4. The 1 primitive triangle with inradius n=1 is (3,4,5). The 4 primitive triangles with n=2 are (5,12,13), (9,10,17), (7,15,20), (6,25,29). The 13 primitive triangles with n=4 are (13,14,15), (15,15,24), (11,25,30), (15,26,37), (10,35,39), (9,40,41), (33,34,65), (25,51,74), (9,75,78), (11,90,97), (21,85,104), (19,153,170), (18,289,305). (Primitive means GCD(a, b, c, n)=1)

Cf. A078644 [Pythagorean triangles with inradius n], A057721 [n^4+3*n^2+1].

Let S(n) be the set of triangles with integer sides a<=b<=c and inradius n. Then:

A120062(n) gives number of triangles in S(n).

A120261(n) gives number of triangles in S(n) with gcd(a, b, c) = 1.

A120252(n) gives number of triangles in S(n) with gcd(a, b, c, n) = 1.

A005408(n) = 2n+1 gives shortest short side a of triangles in S(n).

A120064(n) gives shortest middle side b of triangles in S(n).

A120063(n) gives shortest long side c of triangles in S(n).

A120570(n) gives shortest perimeter of triangles in S(n).

A120572(n) gives smallest area of triangles in S(n).

A058331(n) = 2n^2+1 gives longest short side a of triangles in S(n).

A082044(n) = n^4+2n^2+1 gives longest middle side b of triangles in S(n).

A057721(n) = n^4+3n^2+1 gives longest long side c of triangles in S(n).

A120571(n) = 2n^4+6n^2+4 gives longest perimeter of triangles in S(n).

A120573(n) = gives largest area of triangles in S(n).

Cf. A120252 [primitive triangles with integer inradius], A120063 [minimum of longest sides], A057721 [maximum of longest sides], A120064 [minimum of middle sides], A082044 [maximum of middle sides], A005408 [minimum of shortest sides], A058331 [maximum of shortest sides], A007237 [number of triangles with integer sides and area = n times perimeter].

Sequence in context: A019382 A125146 A051900 this_sequence A081769 A101864 A022138

Adjacent sequences: A120059 A120060 A120061 this_sequence A120063 A120064 A120065

nonn

Hugo Pfoertner (hugo(AT)pfoertner.org), Jun 11 2006

More terms from Graeme McRae (g_m(AT)mcraefamily.com) and Hugo Pfoertner (hugo(AT)pfoertner.org), Jun 12 2006