%I #37 Sep 15 2024 08:55:43

%S 1,1,1,3,-2,11,-29,21,-305,-764,-3761,-26829,20827,-1044667,7336774,

%T -34981779,829881529,4656917815,116074261249,2133710863224,

%U 4261714316929,871401830149817,-15861891538169783,387559539627947379,-20207945101587735626,-195471056819748264101

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

%C 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).

%H Reinhard Zumkeller, <a href="/A178384/b178384.txt">Table of n, a(n) for n = 0..150</a>

%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 LMFDB, <a href="http://www.lmfdb.org/EllipticCurve/Q/91/a/1">Elliptic Curve 91.a1 (Cremona label 91a1)</a>

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

%F From _Michael Somos_, Jan 11 2015: (Start)

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

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

%F 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)

%t 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 *)

%o (Haskell)

%o a178384 n = a178384_list !! n

%o a178384_list = [1, 1, 1, 3] ++

%o zipWith div (foldr1 (zipWith subtract) (map b [1..2])) a178384_list

%o where b i = zipWith (*) (drop i a178384_list) (drop (4-i) a178384_list)

%o -- _Reinhard Zumkeller_, Sep 15 2014

%o (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

%o (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

%o (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 */

%Y Cf. A006720, A247368.

%K easy,sign

%O 0,4

%A _Paul Barry_, May 26 2010