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A006769
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Elliptic divisibility sequence associated with elliptic curve "37a1": y^2 + y = x^3 - x and multiples of the point (0,0).
(Formerly M0157)
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12
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0, 1, 1, -1, 1, 2, -1, -3, -5, 7, -4, -23, 29, 59, 129, -314, -65, 1529, -3689, -8209, -16264, 83313, 113689, -620297, 2382785, 7869898, 7001471, -126742987, -398035821, 1687054711, -7911171596, -47301104551, 43244638645
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OFFSET
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0,6
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COMMENTS
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This sequence has a recursion same as the Somos-4 sequence recursion.
The recurrence formulas in [Kimberling, p. 16] are missing square and cube exponents. - Michael Somos, Jul 07 2014
This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = -1, z = 1.
This sequence and its two subsequences with even/odd indices satisfy the Somos-4 recursion.
The even subsequence is A051138, here called r[ ]. The odd subsequence is the classical Somos-4 A006720, here called s[ ].
These two subsequences interleaved as follows, recover the original sequence which is now: r[0], s[2], r[1], -s[3], r[2], s[4], r[3], -s[5], ..., all Somos-4 s[ ] with odd index with a minus sign. (End)
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REFERENCES
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G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; pp. 11 and 164.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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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) for all n in Z except n=4.
a(n) = (-a(n-1) * a(n-4) - a(n-2) * a(n-3)) / a(n-5) for all n in Z except n=5.
a(-n) = -a(n) for all n in Z.
a(2*n + 1) = a(n+2) * a(n)^3 - a(n-1) * a(n+1)^3, a(2*n) = a(n+2) * a(n) * a(n-1)^2 - a(n) * a(n-2) * a(n+1)^2 for all n in Z.
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MATHEMATICA
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a[n_] := If[n < 0, -a[-n], If[n == 0, 0, ClearAll[an]; an[_] = 1; an[3] = -1; For[k = 5, k <= n, k++, an[k] = (an[k-1]*an[k-3] + an[k-2]^2)/an[k-4]]; an[n]]]; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Dec 14 2011, after first Pari program *)
Join[{0}, RecurrenceTable[{a[1]==a[2]==1, a[3]==-1, a[4]==1, a[n]==(a[n-1] a[n-3]+ a[n-2]^2)/a[n-4]}, a, {n, 40}]] (* Harvey P. Dale, May 04 2018 *)
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PROG
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(PARI) {a(n) = my(an); if( n<0, -a(-n), if( n==0, 0, an = vector( max(3, n), i, 1); an[3] = -1; for( k=5, n, an[k] = (an[k-1] * an[k-3] + an[k-2]^2) / an[k-4]); an[n]))};
(PARI) {a(n) = my(an); if( n<0, -a(-n), if( n==0, 0, an = Vec((-1 - 2*x + sqrt(1 + 4*x - 4*x^3 + O(x^n))) / (2 * x^2)); matdet( matrix((n-1)\2, (n-1)\2, i, j, if(i + j - 1 - n%2<0, 0, an[i + j -n%2])))))};
(PARI) {a(n) = my(E, z); E = ellinit([0, 0, -1, -1, 0]); z = ellpointtoz(E, [0, 0]); round( ellsigma(E, n*z) / ellsigma(E, z)^(n^2))}; /* Michael Somos, Oct 22 2004 */
(PARI) {a(n) = sign(n) * subst( elldivpol( ellinit([0, 0, -1, -1, 0]), abs(n)), x, 0)}; /* Michael Somos, Dec 16 2014 */
(Haskell)
a006769 n = a050512_list !! n
a006769_list = 0 : 1 : 1 : (-1) : 1 : zipWith div (zipWith (+) (zipWith (*)
(drop 4 a006769_list) (drop 2 a006769_list))
(map (^ 2) (drop 3 a006769_list))) (tail a006769_list)
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CROSSREFS
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KEYWORD
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sign,easy,nice
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AUTHOR
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STATUS
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approved
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