A120062 Number of triangles with integer sides a <= b <= c having integer inradius n.

1, 5, 13, 18, 15, 45, 24, 45, 51, 52, 26, 139, 31, 80, 110, 89, 33, 184, 34, 145, 185, 103, 42, 312, 65, 96, 140, 225, 36, 379, 46, 169, 211, 116, 173, 498, 38, 123, 210, 328, 44, 560, 60, 280, 382, 134, 64, 592, 116, 228, 230, 271, 47, 452, 229, 510, 276, 134, 54

Offset: 1

Author

Hugo Pfoertner, Jun 11 2006

Keywords

• nonn
• look

Comments

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 + 2*n^2 + 1, n^4 + 3*n^2 + 1) has inradius n. - David W. Wilson, Jun 17 2006

Previous name was "Number of triangles with integer sides a<=bA362669); so, now effectively, a(10) = 52. - Bernard Schott, Apr 24 2023

Examples

			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.)
		

Links

• David W. Wilson, Table of n, a(n) for n = 1..10000
• Alan F. Beardon and Paul Stephenson, The Heron parameters of a triangle, Mathematical Gazette May 8, 2014.
• Frank M Jackson, Mathematica program
• Thomas Mautsch, Additional terms

Crossrefs

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].

Cf. A362669, A362670.

Programs

Mathematica

(* See link above. *)

Formula

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

a(n) = Sum_{k|n} A120252(k).

Extensions

More terms from Graeme McRae and Hugo Pfoertner, Jun 12 2006

Name corrected by Bernard Schott, Apr 24 2023