A178081 A (1,1) Somos-4 sequence.

0, 1, 1, -1, 2, 3, 1, -11, -16, 35, -129, -299, -386, 3977, 8063, -42489, 269344, 1000009, 3727745, -47166649, -123526014, 1764203419, -18228952703, -113727892147, -1065812586544, 18344075481339, 52130069331199, -2470319425874195

Offset: 0

Author

Paul Barry, May 19 2010

Keywords

• easy
• sign

Comments

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

From Paul Barry, May 31 2010: (Start)

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).

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)

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

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

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..215
• C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
• LMFDB, Elliptic Curve 79.a1 (Cremona label 79a1)

Crossrefs

Cf. A174017.

Programs

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

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

Mathematica

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 *)

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 */

SageMath

@CachedFunction
def a(n): # a = A178081
    if (n<5): return (0, 1, 1, -1, 2)[n]
    else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Feb 16 2023

Formula

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

a(-n) = -a(n) for all n in Z. - Michael Somos, Aug 06 2014

Extensions

a(0)=0 prepended by Michael Somos, Aug 06 2014

a(1)=1 added also by Michael Somos, Feb 15 2023