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A206334 Numbers n such that there is a triangle with area n, side n, and the other two sides rational.

%I A206334 #34 Feb 16 2025 08:33:16
%S A206334 3,5,7,10,12,15,16,18,19,23,25,26,27,28,29,30,33,34,36,38,39,40,41,42,
%T A206334 43,44,46,47,51,52,55,57,58,59,62,63,64,65,67,68,69,70,71,72,74,75,76,
%U A206334 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
%N A206334 Numbers n such that there is a triangle with area n, side n, and the other two sides rational.
%C A206334 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.
%C A206334 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.
%C A206334 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.
%C A206334 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.]
%C A206334 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.
%H A206334 James R. Buddenhagen, <a href="/A206334/a206334.txt">Table of triangles up to n = 145</a>
%H A206334 Dragan K and Rita the dog, <a href="http://answers.yahoo.com/question/index;_ylt=As9iUvdShLradeXMqxZh.53sy6IX;_ylv=3?qid=20120127091629AACRNuT">Question and answer</a> [broken link]
%H A206334 Ian Connell, <a href="https://web.archive.org/web/20010911012433/http://www.math.mcgill.ca/connell/">APECS elliptic curve software</a> (which runs under old versions of Maple).
%H A206334 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>.
%e A206334 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.
%Y A206334 Cf. A112087, A054602, A155965, and A206351 (subsequences, see comments).
%K A206334 nonn
%O A206334 1,1
%A A206334 _James R. Buddenhagen_, Feb 06 2012