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%I #13 Sep 08 2022 08:45:54
%S 1,1,-1,-4,-3,19,67,-40,-1243,-4299,25627,334324,627929,-29742841,
%T -372632409,1946165680,128948361769,1488182579081,-52394610324649,
%U -2333568937567764,-5642424912729707,3857844273728205019
%N A (1,1) Somos-4 sequence associated to the elliptic curve E: y^2 - x*y - y = x^3 + x^2 + x.
%C a(n) is (-1)^C(n,2) times the Hankel transform of the sequence with g.f.
%C 1/(1-x^2/(1-x^2/(1-4x^2/(1+(3/16)x^2/(1-(76/9)x^2/(1-(201/361)x^2/(1-... where
%C 1,4,-3/16,76/9,201/361,... are the x-coordinates of the multiples of z=(0,0)
%C on E:y^2-xy-y=x^3+x^2+x.
%H G. C. Greubel, <a href="/A178628/b178628.txt">Table of n, a(n) for n = 0..155</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1912.01126">Riordan arrays, the A-matrix, and Somos 4 sequences</a>, arXiv:1912.01126 [math.CO], 2019.
%F a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), n>3.
%t RecurrenceTable[{a[n] == (a[n-1]*a[n-3] +a[n-2]^2)/a[n-4], a[0] == 1, a[1] == 1, a[2] == -1, a[3] == -4}, a, {n,0,30}] (* _G. C. Greubel_, Sep 18 2018 *)
%o (PARI) a(n)=local(E,z);E=ellinit([ -1,1,-1,1,0]);z=ellpointtoz(E,[0,0]); round(ellsigma(E,n*z)/ellsigma(E,z)^(n^2))
%o (PARI) m=30; v=concat([1,1,-1,-4], vector(m-4)); for(n=5, m, v[n] = ( v[n-1]*v[n-3] +v[n-2]^2)/v[n-4]); v \\ _G. C. Greubel_, Sep 18 2018
%o (Magma) I:=[1,1,-1,-4]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..30]]; // _G. C. Greubel_, Sep 18 2018
%K easy,sign
%O 0,4
%A _Paul Barry_, May 31 2010
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