This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A386980 #10 Aug 19 2025 22:42:17 %S A386980 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, %T A386980 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, %U A386980 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 %N A386980 Number of acute Heronian triangles with integer inradius n. %C A386980 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. %C A386980 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). %C A386980 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. %C A386980 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. %H A386980 Alan F. Beardon and Paul Stephenson, <a href="https://www.researchgate.net/publication/304216979_The_Heron_parameters_of_a_triangle">The Heron parameters of a triangle</a>, Mathematical Gazette May 8, 2014. %H A386980 Frank M Jackson, <a href="/A386980/a386980_2.txt">Mathematica program</a> %e A386980 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). %t A386980 (* See link above. *) %Y A386980 Cf. A078644, A120062, A386981. %K A386980 nonn %O A386980 1,6 %A A386980 _Frank M Jackson_, Aug 11 2025