A223941 Areas of primitive Heron triangles with two rational triangle medians.

420, 55440, 2042040, 23931600, 75698280, 142334216640, 1877686881840, 185643608470320, 2137147184560080

Offset: 1

Author

Arkadiusz Wesolowski, Mar 29 2013

Keywords

• hard
• more
• nonn

Comments

All terms are divisible by 30.

It is not certain whether other values lie between those given. - Peter Luschny and Andrey Zabolotskiy, Apr 08 2024

Links

• Ralph H. Buchholz and Randall L. Rathbun, An infinite set of Heron triangles with two rational medians, The American Mathematical Monthly, Vol. 104, No. 2 (Feb., 1997), pp. 107-115.
• Andrew N. W. Hone, Heron Triangles and the Hunt for Unicorns, Math. Intelligencer (2024); arXiv:2401.05581 [math.NT], 2024.
• Eric Weisstein's World of Mathematics, Heronian Triangle

Crossrefs

Cf. A181928.

A360537 is a subsequence.

Programs

Mathematica

(*Brute-force search*)lst = {}; Do[s = (a + b + c)/2; d = Sqrt[s*(s - a)*(s - b)*(s - c)]; If[IntegerQ[d] && Divisible[d, 30] && d > 0, p = {{a, c, b}, {b, c, a}}; t = 0; Do[m = 1/2*Sqrt[2*p[[n, 1]]^2 + 2*p[[n, 2]]^2 - p[[n, 3]]^2]; If[MatchQ[m, _Rational] || IntegerQ[m], t++, Break[]], {n, 2}]; If[t == 2, AppendTo[lst, d]]], {a, 73}, {b, 51}, {c, 26}]; lst

Extensions

a(7)-a(9) from Hone (2024) added by Andrey Zabolotskiy, Apr 06 2024