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

%I A120062 #39 Aug 11 2025 16:14:25
%S A120062 1,5,13,18,15,45,24,45,51,52,26,139,31,80,110,89,33,184,34,145,185,
%T A120062 103,42,312,65,96,140,225,36,379,46,169,211,116,173,498,38,123,210,
%U A120062 328,44,560,60,280,382,134,64,592,116,228,230,271,47,452,229,510,276,134,54
%N A120062 Number of triangles with integer sides a <= b <= c having integer inradius n.
%C A120062 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.
%C A120062 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
%C A120062 Previous name was "Number of triangles with integer sides a<=b<c having integer inradius n." With this previous definition, a(10) should be 51 and not 52, because there exists an isosceles triangle with sides (30, 39, 39) and inradius r = 10 (see A362669); so, now effectively, a(10) = 52. - _Bernard Schott_, Apr 24 2023
%H A120062 David W. Wilson, <a href="/A120062/b120062.txt">Table of n, a(n) for n = 1..10000</a>
%H A120062 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 A120062 Frank M Jackson, <a href="/A120062/a120062.txt">Mathematica program</a>
%H A120062 Thomas Mautsch, <a href="http://groups.google.com/group/de.sci.mathematik/msg/88541089a50e7b68">Additional terms</a>
%F A120062 The even-numbered terms are given by a(2*n)=A007237(n).
%F A120062 a(n) = Sum_{k|n} A120252(k).
%e A120062 a(1)=1: {3,4,5} is the only triangle with integer sides and inradius 1.
%e A120062 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.
%e A120062 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.)
%t A120062 (* See link above. *)
%Y A120062 Cf. A078644 [Pythagorean triangles with inradius n], A057721 [n^4+3*n^2+1].
%Y A120062 Let S(n) be the set of triangles with integer sides a<=b<=c and inradius n. Then:
%Y A120062 A120062(n) gives number of triangles in S(n).
%Y A120062 A120261(n) gives number of triangles in S(n) with gcd(a, b, c) = 1.
%Y A120062 A120252(n) gives number of triangles in S(n) with gcd(a, b, c, n) = 1.
%Y A120062 A005408(n) = 2n+1 gives shortest short side a of triangles in S(n).
%Y A120062 A120064(n) gives shortest middle side b of triangles in S(n).
%Y A120062 A120063(n) gives shortest long side c of triangles in S(n).
%Y A120062 A120570(n) gives shortest perimeter of triangles in S(n).
%Y A120062 A120572(n) gives smallest area of triangles in S(n).
%Y A120062 A058331(n) = 2n^2+1 gives longest short side a of triangles in S(n).
%Y A120062 A082044(n) = n^4+2n^2+1 gives longest middle side b of triangles in S(n).
%Y A120062 A057721(n) = n^4+3n^2+1 gives longest long side c of triangles in S(n).
%Y A120062 A120571(n) = 2n^4+6n^2+4 gives longest perimeter of triangles in S(n).
%Y A120062 A120573(n) = gives largest area of triangles in S(n).
%Y A120062 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].
%Y A120062 Cf. A362669, A362670.
%K A120062 nonn,look
%O A120062 1,2
%A A120062 _Hugo Pfoertner_, Jun 11 2006
%E A120062 More terms from _Graeme McRae_ and _Hugo Pfoertner_, Jun 12 2006
%E A120062 Name corrected by _Bernard Schott_, Apr 24 2023