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A178628 A (1,1) Somos-4 sequence associated to the elliptic curve E: y^2 - x*y - y = x^3 + x^2 + x.

%I A178628 #23 Jul 05 2024 16:07:02
%S A178628 1,1,-1,-4,-3,19,67,-40,-1243,-4299,25627,334324,627929,-29742841,
%T A178628 -372632409,1946165680,128948361769,1488182579081,-52394610324649,
%U A178628 -2333568937567764,-5642424912729707,3857844273728205019
%N A178628 A (1,1) Somos-4 sequence associated to the elliptic curve E: y^2 - x*y - y = x^3 + x^2 + x.
%C A178628 a(n) is (-1)^C(n,2) times the Hankel transform of the sequence with g.f.
%C A178628 1/(1-x^2/(1-x^2/(1-4x^2/(1+(3/16)x^2/(1-(76/9)x^2/(1-(201/361)x^2/(1-... where
%C A178628 1,4,-3/16,76/9,201/361,... are the x-coordinates of the multiples of z=(0,0)
%C A178628 on E:y^2-xy-y=x^3+x^2+x.
%H A178628 G. C. Greubel, <a href="/A178628/b178628.txt">Table of n, a(n) for n = 1..160</a>
%H A178628 Paul Barry, <a href="https://arxiv.org/abs/1912.01126">Riordan arrays, the A-matrix, and Somos 4 sequences</a>, arXiv:1912.01126 [math.CO], 2019.
%F A178628 a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), n>4.
%F A178628 a(n) = -a(-n). a(n) = (-a(n-1)*a(n-4) +4*a(n-2)*a(n-3))/a(n-5) for all n in Z except n=5. - _Michael Somos_, Jul 05 2024
%t A178628 RecurrenceTable[{a[n] == (a[n-1]*a[n-3] +a[n-2]^2)/a[n-4], a[1] == 1, a[2] == 1, a[3] == -1, a[4] == -4}, a, {n,1,30}] (* _G. C. Greubel_, Sep 18 2018 *)
%o A178628 (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))
%o A178628 (PARI) m=30; v=concat([1,1,-1,-4], 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 A178628 (PARI) {a(n) = subst(elldivpol(ellinit([-1, 1, -1, 1, 0]), n), x ,0)}; /* _Michael Somos_, Jul 05 2024 */
%o A178628 (Magma) I:=[1,1,-1,-4]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..30]]; // _G. C. Greubel_, Sep 18 2018
%o A178628 (SageMath)
%o A178628 @CachedFunction
%o A178628 def a(n): # a = A178628
%o A178628     if n<5: return (0,1,1,-1,-4)[n]
%o A178628     else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)
%o A178628 [a(n) for n in range(1,41)] # _G. C. Greubel_, Jul 05 2024
%Y A178628 Cf. A006720.
%Y A178628 (p, q) Somos-4 sequences: A171422, A174168, A174170, A174404, A174809, A174811, A174882, A178075, A178077, A178081, A178079, A178376, A178377, A178384, A178417, A178418, A178621, A178622, A178624, A178625, A178627, A178628, A178644, A184019, A184121, A188313, A188315, A352625.
%K A178628 easy,sign
%O A178628 1,4
%A A178628 _Paul Barry_, May 31 2010
%E A178628 Offset changed to 0. - _Michael Somos_, Jul 05 2024