A386980 Number of acute Heronian triangles with integer inradius n.

0, 0, 1, 1, 0, 4, 0, 2, 2, 2, 0, 6, 0, 1, 4, 3, 0, 8, 0, 6, 7, 2, 0, 17, 1, 0, 2, 8, 0, 14, 0, 3, 6, 1, 4, 17, 0, 0, 4, 12, 0, 27, 0, 4, 13, 1, 0, 27, 1, 4, 2, 4, 0, 13, 5, 14, 2, 0, 0, 32, 0, 0, 14, 4, 3, 18, 0, 5, 3, 15, 0, 41, 0, 0, 10, 4, 7, 16, 0, 18, 3, 0, 0, 60, 2, 0, 2, 18, 0, 39, 9

Offset

1,6

Comments

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

The only Heronian triangle with inradius 1 is the right triangle (3, 4, 5). Also, it has been proved that other than n = 3, all acute Heronian triangles have no prime inradii. For n = 3, the Heronian triangle has sides (10, 10, 12).

Empirically, it appears that the remaining occurrences of zero counts (other than 1 and the primes excluding 3) are inradii of the form 2p where p is in the set 13, 19, 29 and all other primes > 29.

The number of right integer triangles with inradius n is given by A078644, the number of obtuse Heronian triangles with inradius n is given by A386981 and the total number of Heronian triangles with inradius n is given by A120062.

Links

Table of n, a(n) for n=1..91.

Alan F. Beardon and Paul Stephenson, The Heron parameters of a triangle, Mathematical Gazette May 8, 2014.

Frank M Jackson, Mathematica program

Example

a(6) = 4, and the 4 acute Heronian triangles with inradius 6 have sides (15, 34, 35), (17, 25, 28), (17, 25, 26), (20, 20, 24).

Mathematica

(* See link above. *)

Crossrefs

Cf. A078644, A120062, A386981.

Sequence in context: A090538 A371860 A320157 * A320479 A218769 A180850

Adjacent sequences: A386977 A386978 A386979 * A386981 A386982 A386983

Keyword

nonn

Author

Frank M Jackson, Aug 11 2025

Status

approved