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A206334 Numbers n such that there is a triangle with area n, side n, and the other two sides rational. 3
3, 5, 7, 10, 12, 15, 16, 18, 19, 23, 25, 26, 27, 28, 29, 30, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 46, 47, 51, 52, 55, 57, 58, 59, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 80, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 95, 96, 97, 103, 104, 105, 106, 107, 109, 115, 119, 122, 123, 124, 125, 126 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..75.

James R. Buddenhagen, Table of triangles up to n=145

Dragan K and Rita the dog, Question and answer

Ian Connell, APECS elliptic curve software (which runs under old versions of Maple).

Eric Weisstein, MathWorld: 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

Cf. A112087, A054602, A155965, and A206351 (subsequences, see comments).

Sequence in context: A033035 A047390 A184653 * A184741 A020959 A175312

Adjacent sequences:  A206331 A206332 A206333 * A206335 A206336 A206337

KEYWORD

nonn

AUTHOR

James R. Buddenhagen, Feb 06 2012

STATUS

approved

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Last modified April 18 11:01 EDT 2014. Contains 240707 sequences.