|
%I #15 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 y^2 + x*y + y = x^3 + x^2 + x.
%C Hankel transform of the sequence with g.f. 1/(1-x^2/(1-x^2/(1-4x^2/(1+(3/16)x^2/(1-(76/9)x^2/(1-.... where 1,4,-3/16,76/9,... are the x-coordinates of the multiples of (0,0).
%H G. C. Greubel, <a href="/A178417/b178417.txt">Table of n, a(n) for n = 1..156</a> (offset adapted by _Georg Fischer_, Jan 31 2019).
%F a(n) = (-a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), n>3.
%F a(n) = -(-1)^n*a(-n) for all n in Z. - _Michael Somos_, Sep 17 2018
%e G.f. = x + x^2 + x^3 + 4*x^4 - 3*x^5 + 19*x^6 - 67*x^7 + ... - _Michael Somos_, Sep 17 2018
%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 16 2018 *)
%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 16 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 16 2018
%K easy,sign
%O 1,4
%A _Paul Barry_, May 27 2010
%E Changed offset to 1 by _Michael Somos_, Sep 17 2018
|
