A178624 - OEIS

%I #14 Sep 08 2022 08:45:54

%S 1,1,-3,11,38,249,-2357,8767,496035,-3769372,-299154043,-12064147359,

%T 632926474117,-65604679199921,-6662962874355342,-720710377683595651,

%U 285131375126739646739,5206174703484724719135

%N A (1,3) Somos-4 sequence associated to the elliptic curve E: y^2 + 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. 1/(1-x^2/(1-3x^2/(1+(11/9)x^2/(1-(114/121)x^2/(1+(2739/1444)x^2/(1-... where 3,-11/9,141/121,-2739/1444... are the x-coordinates of the multiples of z=(0,0) on E:y^2+2xy-y=x^3-x.

%H G. C. Greubel, <a href="/A178624/b178624.txt">Table of n, a(n) for n = 1..118</a> (offset adapted by _Georg Fischer_, Jan 31 2019)

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

%F a(n) = -a(-n) for all n in Z. - _Michael Somos_, Sep 17 2018

%e G.f. = x + x^2 - 3*x^3 + 11*x^4 + 38*x^5 + 249*x^6 + ... - _Michael Somos_, Sep 17 2018

%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] == 11}, a, {n, 0, 30}] (* _G. C. Greubel_, Sep 16 2018 *)

%o (PARI) a(n)=local(E,z);E=ellinit([2,0,-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,-3,11], 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_, Sep 16 2018

%o (Magma) I:=[1,1,-3,11]; [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_, Sep 16 2018

%K easy,sign

%O 1,3

%A _Paul Barry_, May 31 2010

%E Corrected by _Paul Barry_, Jun 01 2010

%E Offset changed to 1 by _Michael Somos_, Sep 17 2018