%I #19 Sep 08 2022 08:45:54

%S 1,1,-2,5,13,24,-229,-365,7394,-59449,-572233,-7773360,151071097,

%T -578687351,-87627413666,1616831255645,95461592201461,

%U 5420484489890376,-308006807300556157,18159275356643111275,3018684797307268036418

%N A (1, 2) Somos-4 sequence associated to the elliptic curve E: y^2 + x*y - y = x^3 - x.

%C a(n) is (-1)^C(n,2) times the Hankel transform of the sequence with g.f.

%C 1/(1 - x/(1 - 2x^2/(1 + (5/4)x^2/(1 - (26/25)x^2/(1 + (120/169)x^2/(1 - ... where

%C 2, -5/4, 26/25, -120/169 are the x-coordinates of the multiples of z=(0,0) on E.

%H G. C. Greubel, <a href="/A178621/b178621.txt">Table of n, a(n) for n = 1..142</a> (offset adapted by _Georg Fischer_, Jan 31 2019).

%F a(n) = (a(n-1)*a(n-3) + 2*a(n-2)^2)/a(n-4), n > 4.

%F a(n) = -a(-n), a(n+3)*a(n-2) = -2*a(n+2)*a(n-1) - 5*a(n+1)*a(n) for all n in Z. - _Michael Somos_, Sep 19 2018

%e G.f. = x + x^2 - 2*x^3 + 5*x^4 + 13*x^5 + 24*x^6 - 365*x^7 + ... - _Michael Somos_, Sep 19 2018

%t RecurrenceTable[{a[1]==a[2]==1,a[3]==-2,a[4]==5,a[n]==(a[n-1]a[n-3]+ 2a[n-2]^2)/a[n-4]},a[n],{n,30}] (* _Harvey P. Dale_, Sep 30 2011 *)

%o (PARI) a(n)=local(E, z); E=ellinit([1, 0, -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,-2,5], vector(m-4)); for(n=5, m, v[n] = ( v[n-1]*v[n-3] + 2*v[n-2]^2)/v[n-4]); v \\ _G. C. Greubel_, Sep 18 2018

%o (Magma) I:=[1,1,-2,5]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + 2*Self(n-2)^2)/Self(n-4): n in [1..30]]; // _G. C. Greubel_, Sep 18 2018

%K easy,sign

%O 1,3

%A _Paul Barry_, May 31 2010

%E Changed offset to 1 by _Michael Somos_, Sep 19 2018