OFFSET
1,1
COMMENTS
n>3 is in the sequence just in case the elliptic curve y^2 = 4*x^4 + (n^2+8)*x^2 + 4 has positive rank. Note that (0,2) is on that curve.
n is in the sequence just in case there are positive rational numbers x,y such that x*y>1 and x - 1/x + y - 1/y = n.
The triangle whose sides are [(4*k^6+8*k^5+8*k^4+4*k^3+2*k^2+2*k+1)/((k+1)*k*(2*k^2+2*k+1)), (4*k^6+16*k^5+28*k^4+28*k^3+18*k^2+6*k+1)/((k+1)*k*(2*k^2+2*k+1)), 4*k^2+4*k+4] has area equal to its third side. Hence, starting with the second term, A112087 is a subsequence of the present sequence.
The triangle whose sides are [(k^6+2*k^4+k^2+1)/(k*(k^2+1)), (k^4+3*k^2+1)/(k*(k^2+1)), (k^2+2)*k] has area equal to its third side. Hence, starting with the first positive term, A054602 is a subsequence of the present sequence. [This subsequence found by Dragan K, see second link, below.]
The triangle whose sides are [(k^8+6*k^6+13*k^4+13*k^2+4)/(k*(k^2+2)*(k^2+1)), (k^6+3*k^4+5*k^2+4)/(k*(k^2+2)*(k^2+1)), k*(k^2+4)] has area equal to its third side. Hence A155965 is a subsequence of the present sequence.
LINKS
James R. Buddenhagen, Table of triangles up to n = 145
Dragan K and Rita the dog, Question and answer [broken link]
Ian Connell, APECS elliptic curve software (which runs under old versions of Maple).
Eric Weisstein's World of Mathematics, Heronian Triangle.
EXAMPLE
5 is in the sequence because the triangle with sides (37/6, 13/6, 5) has area 5, one side 5, and the other two sides rational.
CROSSREFS
KEYWORD
nonn
AUTHOR
James R. Buddenhagen, Feb 06 2012
STATUS
approved