A174017 -id:A174017 - cp's OEIS frontend

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

Paul Barry, May 19 2010

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

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

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)

Cf. A174017.

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

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

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 *)

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 */

@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

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

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

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

A174013 Sequence whose Hankel transform is a (1,1) Somos-4 sequence.

Original entry on oeis.org

1, 1, 3, 6, 12, 22, 37, 56, 73, 75, 44, -21, -39, 297, 1751, 5749, 14104, 27136, 38163, 22135, -80421, -369611, -934754, -1637758, -1559395, 2019629, 14766699, 44732254, 94865112, 138114302, 61077521

Offset: 0

Paul Barry, Mar 05 2010

Continued fraction form of g.f. A(x) given by A(x) = 1/(1-x*(1+x)/(1-x/(1+x*A(x)))).
Hankel transform is A174017.
Diagonal sums of the Deleham array [1,1,-1,1,1,-1,1,...] Delta [1,0,0,1,0,0,1,0,0,1,0,...], or A174014.

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 12*x^4 + 22*x^5 + 37*x^6 + ... - _Michael Somos_, Jul 11 2024

Cf. A174014, A174017.

a[ n_] := SeriesCoefficient[2*(1-x)/(1-3*x-x^2 + Sqrt[1-2*x-x^2+6*x^3+5*x^4]), {x, 0, n}]; (* Michael Somos, Jul 11 2024 *)

{a(n) = polcoeff(2*(1-x)/(1-3*x-x^2 + sqrt(1-2*x-x^2+6*x^3+5*x^4 + x*O(x^n))), n)}; /* Michael Somos, Jul 11 2024 */

G.f.: -(1-3*x-x^2-sqrt(1-2*x-x^2+6*x^3+5*x^4))/(2*x*(1-x-x^2)).
G.f.: 1/(1-x*(1+x)/(1-x/(1+x/(1-x*(1+x)/(1-x/(1+x/(1-... (continued fraction).
Conjecture: (n+1)*a(n) +n*a(n-1) +12*(-n+1)*a(n-2) +3*(3*n-8)*a(n-3) +6*(6*n-23)*a(n-4) +11*(-n+2)*a(n-5) +(-49*n+253)*a(n-6) +20*(-n+6)*a(n-7)=0. - R. J. Mathar, Jan 12 2013

cp's OEIS Frontend

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

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