Search: id:A120062
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%I A120062
%S A120062 1,5,13,18,15,45,24,45,51,52,26,139,31,80,110,89,33,184,34,145,185,103,
%T A120062 42,312,65,96,140,225,36,379,46,169,211,116,173,498,38,123,210,328,44,
%U A120062 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+2n^2+1, 
               n^4+3n^2+1) has inradius n. - David W. Wilson, Jun 17 2006
%H A120062 David W. Wilson, <a href="b120062.txt">Table of n, a(n) for n = 1..10000</
               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: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)
%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 Sequence in context: A145040 A125146 A051900 this_sequence A081769 A101864 
               A022138
%Y A120062 Adjacent sequences: A120059 A120060 A120061 this_sequence A120063 A120064 
               A120065
%K A120062 nonn
%O A120062 1,2
%A A120062 Hugo Pfoertner (hugo(AT)pfoertner.org), Jun 11 2006
%E A120062 More terms from Graeme McRae (g_m(AT)mcraefamily.com) and Hugo Pfoertner 
               (hugo(AT)pfoertner.org), Jun 12 2006

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