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A178627
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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).
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1
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0, 1, 1, -1, -2, -1, 5, 9, -8, -41, -61, 241, 770, -271, -8649, -27329, 106768, 651521, 740665, -18425809, -107300098, 399122991, 5615422669, 24055184809, -383354254360, -3943757411849, 9276714153611, 498726356978849
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OFFSET
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0,5
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COMMENTS
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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.
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
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LINKS
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FORMULA
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a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), n>4.
a(n) = -a(-n) for all n in Z.
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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))};
(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
(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
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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