A227003 Number of primitive Heronian triangles with area 6n.

1, 2, 0, 1, 1, 2, 1, 0, 0, 4, 1, 1, 0, 4, 2, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 2, 0, 3, 0, 2, 0, 0, 1, 1, 6, 1, 0, 0, 1, 1, 0, 3, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 0, 0, 4, 4, 0, 0, 0, 3, 0, 0, 0, 0, 1, 3, 0, 1, 0, 15, 0, 0, 0, 0, 0, 3, 2, 1, 0, 1, 0, 0, 0, 3, 2, 0, 1, 2, 0, 0

Offset: 1

Author

Frank M Jackson, Jun 26 2013

Keywords

• nonn

Comments

The Mathematica program captures all primitive Heronian areas up to 540 by searching through integer triangles with a longest side ranging from 3 to at least 484. This upper limit for the longest side is determined by observing that the shortest side of a Heronian triangle is >= 3 and the smallest area of an integer triangle with longest side z and shortest side 3 is generated by the integer triple (3, z-2, z).

Examples

			a(10) = 4 as there are 4 primitive Heronian triangles with area 60. The triples are (10,13,13), (8,15,17), (13,13,24), (6,25,29).
		

Crossrefs

Cf. A051585, A083875.

Programs

Mathematica

nn=540; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s]&&GCD[a, b, c]==1, area2=s(s-a)(s-b)(s-c); If[area2>0&&IntegerQ[Sqrt[area2]], AppendTo[lst, Sqrt[area2]]]], {a, 3, nn}, {b, a}, {c, b}]; lst1=Sort@lst/6; Table[Length@Select[lst1, n==# &], {n, 1, nn/6}] (* using T. D. Noe's program A083875 *)

PARI

a(n)=sum(z=sqrtint(sqrtint(192*n^2)-1)+1,sqrtint(9*(64*n^2+5)\20), sum(y=z\2+1,z, my(t=(y*z)^2-(12*n)^2,x,g=gcd(y,z)); if(issquare(t,&t), (issquare(y^2+z^2-2*t,&x) && gcd(x,g)==1 && x<=y) + (t && issquare(y^2+z^2+2*t,&x) && gcd(x,g)==1 && x<=y), 0))) \\ Charles R Greathouse IV, Jun 27 2013