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

1,2

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).

Links

Table of n, a(n) for n=1..90.

Example

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).

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 *)

Prog

(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

Crossrefs

Cf. A051585, A083875.

Sequence in context: A071466 A155041 A334296 * A307431 A263848 A348157

Adjacent sequences: A227000 A227001 A227002 * A227004 A227005 A227006

Keyword

nonn

Author

Frank M Jackson, Jun 26 2013

Status

approved