The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A120062 Number of triangles with integer sides a<=b
 1, 5, 13, 18, 15, 45, 24, 45, 51, 52, 26, 139, 31, 80, 110, 89, 33, 184, 34, 145, 185, 103, 42, 312, 65, 96, 140, 225, 36, 379, 46, 169, 211, 116, 173, 498, 38, 123, 210, 328, 44, 560, 60, 280, 382, 134, 64, 592, 116, 228, 230, 271, 47, 452, 229, 510, 276, 134, 54 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. 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 LINKS David W. Wilson, Table of n, a(n) for n = 1..10000 Thomas Mautsch, Additional terms FORMULA The even numbered terms are given by a(2*n)=A007237(n). a(n) = sum_{k:k|n} A120252(k) EXAMPLE a(1)=1: {3,4,5} is the only triangle with integer sides and inradius 1. 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. 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) CROSSREFS Cf. A078644 [Pythagorean triangles with inradius n], A057721 [n^4+3*n^2+1]. Let S(n) be the set of triangles with integer sides a<=b<=c and inradius n. Then: A120062(n) gives number of triangles in S(n). A120261(n) gives number of triangles in S(n) with gcd(a, b, c) = 1. A120252(n) gives number of triangles in S(n) with gcd(a, b, c, n) = 1. A005408(n) = 2n+1 gives shortest short side a of triangles in S(n). A120064(n) gives shortest middle side b of triangles in S(n). A120063(n) gives shortest long side c of triangles in S(n). A120570(n) gives shortest perimeter of triangles in S(n). A120572(n) gives smallest area of triangles in S(n). A058331(n) = 2n^2+1 gives longest short side a of triangles in S(n). A082044(n) = n^4+2n^2+1 gives longest middle side b of triangles in S(n). A057721(n) = n^4+3n^2+1 gives longest long side c of triangles in S(n). A120571(n) = 2n^4+6n^2+4 gives longest perimeter of triangles in S(n). A120573(n) = gives largest area of triangles in S(n). 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]. Sequence in context: A051900 A275800 A294136 * A081769 A188030 A101864 Adjacent sequences:  A120059 A120060 A120061 * A120063 A120064 A120065 KEYWORD nonn,look AUTHOR Hugo Pfoertner, Jun 11 2006 EXTENSIONS More terms from Graeme McRae and Hugo Pfoertner, Jun 12 2006 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 10 11:31 EDT 2021. Contains 342845 sequences. (Running on oeis4.)