A386981 Number of obtuse Heronian triangles with integer inradius n.

0, 3, 9, 14, 12, 35, 21, 39, 44, 44, 23, 124, 28, 73, 97, 81, 30, 166, 31, 130, 169, 95, 39, 283, 59, 90, 131, 208, 33, 347, 43, 160, 196, 109, 160, 466, 35, 117, 197, 304, 41, 515, 57, 267, 354, 127, 61, 550, 110, 214, 219, 258, 44, 425, 215, 484, 265, 128, 51, 977, 41, 138, 582, 269, 169, 603, 48, 325, 252, 564, 47, 1058, 65, 133, 445, 341

Offset: 1

Author

Frank M Jackson, Aug 11 2025

Keywords

• nonn
• new

Comments

If a Heronian triangle has an inradius n, and sides (x, y, z), where x <= y <= z, then the triangle is obtuse iff n > (x+y-z)/2.

The only Heronian triangle with inradius 1 is the right triangle (3, 4, 5).

The number of right integer triangles with inradius n is given by A078644, the number of acute Heronian triangles with inradius n is given by A386980 and the total number of Heronian triangles with inradius n is given by A120062.

Examples

			a(2) = 3, and the 3 obtuse Heronian triangles with inradius 2 have sides (6, 25, 29), (7, 15, 20), (9, 10, 17).
		

Links

• Alan F. Beardon and Paul Stephenson, The Heron parameters of a triangle, Mathematical Gazette May 8, 2014.
• Frank M Jackson, Mathematica program

Crossrefs

Cf. A078644, A120062, A386980

Programs

Mathematica

(* See link above. *)