%I #31 Oct 23 2024 00:49:23

%S 0,1,1,2,1,-7,-16,-57,-113,670,3983,23647,140576,-833503,-14871471,

%T -147165662,-2273917871,11396432249,808162720720,14252325989831,

%U 503020937289311,23268424032702,-625775582778294689,-22086170583356766977,-1557994930804790259136,-27620103680757212617727,6783061219100782906098017,547569584492952570186575810

%N A (1, -2) Somos-4 sequence associated to the elliptic curve E: y^2 - 3*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+2x^2/(1+(1/4)x^2/(1-14x^2/(1-(16/49)x^2/(1-... where 0/1, -2/1, -1/4, 14/1, 16/49, ... are the x-coordinates of the multiples of z=(0, 0) on E.

%C This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = 2, z = 1. - _Michael Somos_, Aug 06 2014

%H G. C. Greubel, <a href="/A178622/b178622.txt">Table of n, a(n) for n = 0..155</a>

%H Paul Barry, <a href="https://arxiv.org/abs/1807.05794">Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences</a>, arXiv:1807.05794 [math.CO], 2018.

%H Paul Barry, <a href="https://arxiv.org/abs/1910.00875">Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials</a>, arXiv:1910.00875 [math.CO], 2019.

%H C. Kimberling, <a href="http://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.

%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+5)*a(n) = 2*a(n+4)*a(n+1) - a(n+3)*a(n+2) for all n in Z. - _Michael Somos_, Aug 06 2014

%F a(n) = A242107(2*n) for all n in Z. - _Michael Somos_, Oct 22 2024

%e G.f. = x + x^2 + 2*x^3 + x^4 - 7*x^5 - 16*x^6 - 57*x^7 - ... - _Michael Somos_, Oct 22 2024

%t nxt[{a_,b_,c_,d_}]:={b,c,d,(d*b-2c^2)/a}; Join[{0},Transpose[ NestList[ nxt,{1,1,2,1},30]][[1]]] (* _Harvey P. Dale_, Aug 19 2015 *)

%t Join[{0}, RecurrenceTable[{a[n] == (a[n-1]*a[n-3] -2*a[n-2]^2)/a[n - 4], a[1] == 1, a[2] == 1, a[3] == 2, a[4] == 1}, a, {n, 1, 30}]] (* _G. C. Greubel_, Sep 18 2018 *)

%t a[ n_] := Which[n == 0, 0, n < 0, -a[-n], n < 5, {1, 1, 2, 1}[[n]], True, a[n] = (a[n-1]*a[n-3] - 2*a[n-2]*a[n-2])/a[n-4]]; (* _Michael Somos_, Oct 22 2024 *)

%o (PARI) {a(n) = my(E,z); E=ellinit([3, 0, 1, -1, 0]); z=ellpointtoz(E,[0,0]); -(-1)^n*round(ellsigma(E,n*z)/ellsigma(E,z)^(n^2))};

%o (PARI) m=30; v=concat([0,1,1,2,1], vector(m-5)); for(n=6, 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:=[0, 1, 1, 2, 1]; [n le 5 select I[n] else (Self(n-1)*Self(n-3)-2*Self(n-2)^2)/Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Aug 07 2014

%Y Cf. A242107.

%K easy,sign

%O 0,4

%A _Paul Barry_, May 31 2010

%E Added missing a(0)=0.

%E More terms from _Vincenzo Librandi_, Aug 07 2014