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

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

Offset

1,1

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

Table of n, a(n) for n=1..9.

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

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

Crossrefs

Cf. A181928.

A360537 is a subsequence.

Sequence in context: A135340 A135380 A035846 * A360537 A046167 A059951

Adjacent sequences: A223938 A223939 A223940 * A223942 A223943 A223944

Keyword

hard,more,nonn

Author

Arkadiusz Wesolowski, Mar 29 2013

Extensions

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

Status

approved