A178081 - OEIS

%I #35 Feb 16 2023 17:10:36

%S 0,1,1,-1,2,3,1,-11,-16,35,-129,-299,-386,3977,8063,-42489,269344,

%T 1000009,3727745,-47166649,-123526014,1764203419,-18228952703,

%U -113727892147,-1065812586544,18344075481339,52130069331199,-2470319425874195

%N A (1,1) Somos-4 sequence.

%C Hankel transform of A178080 is a(n+2).

%C From _Paul Barry_, May 31 2010: (Start)

%C The sequence 1,1,-1,2,3,... is associated to the elliptic curve E:y^2+xy-y=x^3+x^2-x (see PARI code below).

%C This is also (-1)^C(n,2) times the Hankel transform of the sequence whose g.f. is 1/(1-x^2/(1-x^2/(1+2x^2/(1-(3/4)x^2/(1+(2/9)x^2/(1-...)))))) where 1, -2, 3/4, -2/9, 33, ... are the x-coordinates of the multiples of z=(0,0) on the elliptic curve E:y^2+xy-y=x^3+x^2-x. (End)

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

%C This is associated with elliptic curve y^2 + xy - y = x^3 + x^2 - 2x (Cremona label 79a1) and multiples of the point (0, 0). - _Michael Somos_, Feb 15 2023

%H G. C. Greubel, <a href="/A178081/b178081.txt">Table of n, a(n) for n = 0..215</a>

%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.

%H LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/79/a/1">Elliptic Curve 79.a1 (Cremona label 79a1)</a>

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

%F a(-n) = -a(n) for all n in Z. - _Michael Somos_, Aug 06 2014

%e G.f. = x + x^2 - x^3 + 2*x^4 + 3*x^5 + x^6 - 11*x^7 - 16*x^8 + 35*x^9 + ...

%t Join[{0},RecurrenceTable[{a[1]==1,a[2]==1,a[3]==-1,a[4]==2,a[n]==(a[n-1]a[n-3]+a[n-2]^2)/a[n-4]},a,{n,30}]] (* _Harvey P. Dale_, Sep 07 2016 *)

%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)) /* _Paul Barry_, May 31 2010 */

%o (Magma) I:=[0,1,1,-1,2]; [n le 5 select I[n] else (Self(n-1)*Self(n-3)+Self(n-2)^2)/Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Aug 06 2014

%o (GAP) a:=[1,1,-1,2];; for n in [5..30] do a[n]:=(a[n-1]*a[n-3]+a[n-2]^2)/a[n-4]; od; a:=Concatenation([0],a); # _Muniru A Asiru_, Sep 23 2018

%o (SageMath)

%o @CachedFunction

%o def a(n): # a = A178081

%o     if (n<5): return (0, 1, 1, -1, 2)[n]

%o     else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)

%o [a(n) for n in range(41)] # _G. C. Greubel_, Feb 16 2023

%Y Cf. A174017.

%K easy,sign

%O 0,5

%A _Paul Barry_, May 19 2010

%E a(0)=0 prepended by _Michael Somos_, Aug 06 2014

%E a(1)=1 added also by _Michael Somos_, Feb 15 2023