A006721 -id:A006721 - cp's OEIS frontend

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

Andrey Zabolotskiy, Feb 10 2023

nonn

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

Cf. A006721, A360381.

This is a subsequence of A223941.

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}]

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)].

A097496 Subsequence of terms of odd index of the Somos-5 sequence.

Original entry on oeis.org

1, 1, 2, 5, 37, 274, 6161, 165713, 9434290, 1013908933, 142844426789, 57760865728994, 23510036246274433, 29636604976524724225, 42939094881045167060258, 139127528853472250968193381, 802780326236562434930932841989, 6388127622103180665094420305200242

Offset: 0

Andrew Hone, Aug 24 2004

nonn

The sequence corresponds to the sequence of points Q+nP on the curve y^2=4*x^3-(121/12)*x+845/216, where Q=(-7/12,-3) and P=(17/12,-1).

Another sequence of points is (2*n - 1)*P on the curve "102a1": y^2 + x * y = x^3 + x^2 - 2 * x where P = [-1, -1]. - Michael Somos, Jul 15 2011

A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.

Cf. A006721, A006720.

a[0]:=1; a[1]:=1;a[2]:=2;a[3]:=5; for n from 1 to 20 do a[n+3]:=(a[n+2]*a[n]+8*a[n+1]^2)/a[n-1] od;

{a(n) = sqrtint( denominator( ellpow( ellinit( [1, 1, 0, -2, 0], 1), [-1, -1], 2*n - 1)[1]))}; /* Michael Somos, Jul 15 2011 */

a(0)=a(1)=1, a(2)=2, a(3)=5; a(n) = (a(n-1)*a(n-3)+8*a(n-2)^2)/a(n-4).
Exact formula: a(n)=D*E^n*sigma(f+n*k)/sigma(k)^(n^2) where sigma is the Weierstrass sigma function associated to the elliptic curve y^2=4*x^3-(121/12)*x+845/216, D=1/sigma(f)=-0.363554228-0.803200610*I, E=sigma(k)*sigma(f)/sigma(f+k)=0.644801269+0.734118205*I, f=integral{infinity}^{-7/12} dx/y = -0.509286773+0.973928783*I, k=integral{17/12}^{infinity} dx/y = 1.018573545 all to 9 d.p.
a(n) = a(1 - n), a(n) = (-8 * a(n-1) * a(n-4) + 57 * a(n-2) * a(n-3)) / a(n-5) for all n in Z. - Michael Somos, Jul 15 2011

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

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