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%I #10 Sep 08 2022 08:45:54
%S 1,1,-3,-10,17,249,541,-19520,-234261,4081751,157040423,-675903030,
%T -304046407637,-11362045786001,1814897653228119,243414885066104960,
%U -23403892390201032679,-11906020446293954889999
%N A (1,3) Somos-4 sequence associated to the elliptic curve E: y^2 + x*y - y = x^3 - x^2 + 2*x.
%C (-1)^binomial(n,2) times the Hankel transform of the sequence with g.f. 1/(1 - x^2/(1 - 3x^2/(1 - (10/9)x^2/(1 - (51/100)x^2/(1 - (2490/289)x^2/(1 - ... where 3, 10/9, 51/100, 2490/289, ... are the x-coordinates of the multiples of (0,0) on E: y^2 + xy - y = x^3 - x^2 + 2x.
%H G. C. Greubel, <a href="/A178644/b178644.txt">Table of n, a(n) for n = 0..119</a>
%F a(n) = (a(n-1)*a(n-3) + 3*a(n-2)^2)/a(n-4), n > 3.
%t RecurrenceTable[{a[n]==(a[n-1]*a[n-3] +3*a[n-2]^2)/a[n-4], a[0]==1, a[1] ==1, a[2]==-3, a[3]==-10}, a, {n, 0, 30}] (* _G. C. Greubel_, Jan 28 2019 *)
%o (PARI) a(n)=local(E,z);E=ellinit([1,-1,-1,2,0]);z=ellpointtoz(E,[0,0]);
%o round(ellsigma(E,n*z)/ellsigma(E,z)^(n^2))
%o (PARI) m=30; v=concat([1,1,-3,-10], vector(m-4)); for(n=5, m, v[n] = ( v[n-1]*v[n-3] + 3*v[n-2]^2)/v[n-4]); v \\ _G. C. Greubel_, Jan 28 2019
%o (Magma) I:=[1, 1,-3,-10]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + 3*Self(n-2)^2)/Self(n-4): n in [1..30]]; // _G. C. Greubel_, Jan 28 2019
%K easy,sign
%O 0,3
%A _Paul Barry_, May 31 2010
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