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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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8
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| A Somos-4 sequence.
a(n+1) is the Hankel transform of A178072. [From Paul Barry (pbarry(AT)wit.ie), May 19 2010]
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REFERENCES
| G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; pp. 11 and 164.
C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
H. W. Braden, V. Z. Enolskii and A. N. W. Hone, Bilinear recurrences and addition formulae for hyperelliptic sigma functions
G. Everest, Elliptic Divisibility Sequences
M. Somos, Number Walls in Combinatorics
Index entries for two-way infinite sequences
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FORMULA
| a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4).
a(-n) = -a(n).
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.
A006720(n) = (-1)^n * a(2n-3).
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MATHEMATICA
| 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}] (* From Jean-François Alcover, Dec 14 2011, after first Pari program *)
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PROG
| (PARI) {a(n) = local(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) = local(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) = local(E, z); E = ellinit( [0, 0, -1, -1, 0]); z = ellpointtoz( E, [0, 0]); round( ellsigma( E, n * z) / sqrt( -ellsigma(E, z) * ellsigma( E, 3 * z) / ellsigma( E, 2 * z)^2)^(n^2))} /* Michael Somos Oct 22 2004 */
(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)
-- Reinhard Zumkeller, Nov 02 2011
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CROSSREFS
| A006720(n) = (-1)^n * a(2n-3).
Squared terms give A028941.
Cf. A050512.
Sequence in context: A161169 A058202 A127201 * A075643 A076074 A135017
Adjacent sequences: A006766 A006767 A006768 * A006770 A006771 A006772
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KEYWORD
| sign,easy,nice
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AUTHOR
| Michael Somos, Jul 16 1999
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