A360537 Areas of primitive Heron triangles with two rational medians from the infinite family based on Somos-5 sequences.

420, 55440, 23931600, 142334216640, 2137147184560080, 4323341954766548553840, 18705358317240372854759881380, 1333577710124626249998068999458413600, 248363720675646323338068819310182950300884320, 4199805494977793853528867974891927438920668319491840

Offset: 1

Author

Andrey Zabolotskiy, Feb 10 2023

Keywords

• nonn

Links

• Andrew N. W. Hone, Heron triangles with two rational medians and Somos-5 sequences, European Journal of Mathematics, 8 (2022), 1424-1486; arXiv:2107.03197 [math.NT], 2021-2022.
• Andrew N. W. Hone, Heron Triangles and the Hunt for Unicorns, Math. Intelligencer (2024); arXiv:2401.05581 [math.NT], 2024.
• Wikipedia, Heronian triangle

Crossrefs

Cf. A006721, A360381.

This is a subsequence of A223941.

Programs

Mathematica

t[1|3|4] = 1; t[2] = -1; t[5] = -7;
s[-2|-1|0|1|2] = 1;
Do[f[n_] := f[n] = (f[n-1] f[n-4] + f[n-2] f[n-3]) / f[n-5], {f, {t, s}}];
a[n_] := Abs@Product[f[n] f[n+1] f[n+2]^2 f[n+3] f[n+4], {f, {s, t}}];
Table[a[n], {n, 10}]

Formula

a(n) = |S(n)*S(n+1)*S(n+2)^2*S(n+3)*S(n+4)*T(n)*T(n+1)*T(n+2)^2*T(n+3)*T(n+4)|, where S(n) = A006721(n+2) and T(n) = A360381(n) [Hone, Eq. (1.21)].