A358210 Congruent number sequence starting from the Pythagorean triple (3,4,5).

6, 15, 34, 353, 175234, 9045146753, 121609715057619333634, 4138643330264389621194448797227488932353, 27728719906622802548355602700962556264398170527494726660553210068191276023007234

Offset: 1

Author

Gerry Martens, Nov 04 2022

Keywords

• nonn

Examples

			Starting with the triple (3,4,5) and choosing the b side we obtain by the recurrence the right triangles: (15/2, 4, 17/2), (136/15, 15/2, 353/30), (5295/136, 272/15, 87617/2040), (47663648/79425, 79425/136, 9045146753/10801800), ...
So a(4) = (5295/136) * (272/15) / 2 = 353.
		

Links

• G. Jacob Martens, Rational right triangles and the Congruent Number Problem, arXiv:2112.09553 [math.GM], 2021, see section 8 The unseen recurrence, equations (79,80).

Crossrefs

Cf. A081465 (numerators of hypotenuses).

Programs

Mathematica

nxt[{n_, p_, q_}] := Module[{n1 = Sqrt[p^4 + 4 n^2 q^4], p1 = p Sqrt[p^4 + 4 n^2 q^4], q1 = q^2 n},
  a = p1/q1; b = 2 n1 q1/p1; c = Sqrt[p1^4 + 4 n1^2 q1^4]/(p1 q1);
  Return [{ a b/2, Numerator[b], Denominator[b]}];]
l = NestList[nxt, {6, 3, 1}, 8] ;
l[[All, 1]]