A178628 A (1,1) Somos-4 sequence associated to the elliptic curve E: y^2 - x*y - y = x^3 + x^2 + x.
1, 1, -1, -4, -3, 19, 67, -40, -1243, -4299, 25627, 334324, 627929, -29742841, -372632409, 1946165680, 128948361769, 1488182579081, -52394610324649, -2333568937567764, -5642424912729707, 3857844273728205019
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..160
- Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.
Crossrefs
Programs
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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
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Mathematica
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 *) -
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))
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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
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PARI
{a(n) = subst(elldivpol(ellinit([-1, 1, -1, 1, 0]), n), x ,0)}; /* Michael Somos, Jul 05 2024 */ -
SageMath
@CachedFunction def a(n): # a = A178628 if n<5: return (0,1,1,-1,-4)[n] else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4) [a(n) for n in range(1,41)] # G. C. Greubel, Jul 05 2024
Formula
a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), n>4.
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
Extensions
Offset changed to 0. - Michael Somos, Jul 05 2024
Comments