A178384 A (-1,1) Somos-4 sequence associated with the elliptic curve y^2 + y = x^3 + x.

1, 1, 1, 3, -2, 11, -29, 21, -305, -764, -3761, -26829, 20827, -1044667, 7336774, -34981779, 829881529, 4656917815, 116074261249, 2133710863224, 4261714316929, 871401830149817, -15861891538169783, 387559539627947379, -20207945101587735626, -195471056819748264101

Offset: 0

Author

Paul Barry, May 26 2010

Keywords

• easy
• sign

Comments

Hankel transform of the sequence with g.f. 1/(1-x^2/(1-x^2/(1-3x^2/(1+(2/9)x^2/(1-(33/4)x^2/(1-...)))))) where 1,3,-2/9,33/4,... are the x-coordinates of the multiples of (0,0).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..150
• Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
• LMFDB, Elliptic Curve 91.a1 (Cremona label 91a1)

Crossrefs

Cf. A006720, A247368.

Programs

Haskell

a178384 n = a178384_list !! n
a178384_list = [1, 1, 1, 3] ++
   zipWith div (foldr1 (zipWith subtract) (map b [1..2])) a178384_list
   where b i = zipWith (*) (drop i a178384_list) (drop (4-i) a178384_list)
-- Reinhard Zumkeller, Sep 15 2014

Magma

I:=[1,1,1,3]; [n le 4 select I[n] else (-Self(n-1)*Self(n-3)+Self(n-2)^2)/Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 24 2015

Mathematica

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] == 3}, a, {n, 0, 30}] (* G. C. Greubel, Sep 18 2018 *)

PARI

m=30; v=concat([1,1,1,3], 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 18 2018

PARI

{a(n) = n++; sign(n) * (-1)^(n\2+n+1) * subst( elldivpol( ellinit([0, 0, 1, 1, 0]), abs(n)), x, 0)}; /* Michael Somos, Sep 14 2024 */

Formula

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

From Michael Somos, Jan 11 2015: (Start)

a(n) = (-1)^n * a(-2-n) for all n in Z.

0 = a(n)*a(n+4) + a(n+1)*a(n+3) - a(n+2)*a(n+2) for all n in Z.

0 = a(n)*a(n+5) + a(n+1)*a(n+4) - 3*a(n+2)*a(n+3) for all n in Z. (End)