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A178627 A (1,1) Somos-4 sequence associated to the elliptic curve E: y^2 + x*y - y = x^3 - x^2 + x and point (0,0).

%I A178627 #29 Jan 05 2025 19:51:39
%S A178627 0,1,1,-1,-2,-1,5,9,-8,-41,-61,241,770,-271,-8649,-27329,106768,
%T A178627 651521,740665,-18425809,-107300098,399122991,5615422669,24055184809,
%U A178627 -383354254360,-3943757411849,9276714153611,498726356978849
%N A178627 A (1,1) Somos-4 sequence associated to the elliptic curve E: y^2 + x*y - y = x^3 - x^2 + x and point (0,0).
%C A178627 a(n) is (-1)^C(n,2) times the Hankel transform of the sequence with g.f. 1/(1-x^2/(1-x^2/(1-2x^2/(1-(1/4)x^2/(1-10x^2/(1-(9/25)x^2/(1-...))))))) where 1,2,-1/4,10,9/25,... are the x-coordinates of the multiples of z=(0,0) on E:y^2+xy-y=x^3-x^2+x.
%C A178627 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
%H A178627 G. C. Greubel, <a href="/A178627/b178627.txt">Table of n, a(n) for n = 0..155</a>
%H A178627 Paul Barry, <a href="https://arxiv.org/abs/1807.05794">Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences</a>, arXiv:1807.05794 [math.CO], 2018.
%H A178627 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 A178627 C. Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.
%F A178627 a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), n>4.
%F A178627 a(n) = -a(-n) for all n in Z.
%e A178627 G.f. = x + x^2 - x^3 - 2*x^4 - x^5 + 5*x^6 + 9*x^7 - 8*x^8 + ... - _Michael Somos_, Sep 19 2018
%t A178627 Join[{0},RecurrenceTable[{a[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_, Dec 19 2015 *)
%o A178627 (PARI) {a(n) = my(E, z); E = ellinit( [1, -1, -1, 1, 0]); z = ellpointtoz( E, [0, 0]); round( ellsigma( E, n*z) / ellsigma( E, z)^(n^2))};
%o A178627 (PARI) m=30; v=concat([0,1,1,-1,-2], vector(m-5)); for(n=6, m, v[n] = ( -v[n-1]*v[n-3] - 2*v[n-2]^2)/v[n-4]); v \\ _G. C. Greubel_, Sep 18 2018
%o A178627 (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
%K A178627 easy,sign
%O A178627 0,5
%A A178627 _Paul Barry_, May 31 2010
%E A178627 Added missing a(0) = 0. - _Michael Somos_, Aug 06 2014