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A001999
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a(n+1) = a(n)(a(n)^2 - 3).
(Formerly M3055 N1239)
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11
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OFFSET
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0,1
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COMMENTS
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The next terms in the sequence contain 102 and 305 digits. [Harvey P. Dale, June 09 2011]
From Peter Bala, Nov 13 2012: (Start)
The present sequence is the case x = 3 of the following general remarks. For other cases see A219160 (x = 4), A219161 (x = 5) and A112845 (x = 6).
Let x > 2 and let alpha := {x + sqrt(x^2 - 4)}/2. Define a sequence a(n) (which depends on x) by setting a(n) = alpha^(3^n) + (1/alpha)^(3^n). Then it is easy to verify that the sequence a(n) satisfies the recurrence equation a(n+1) = a(n)^3 - 3*a(n) with the initial condition a(0) = x.
We have the following identity, valid for x > 2: sqrt((x + 2)/(x - 2)) = (1 + 2/(x-1))*sqrt((y + 2)/(y - 2)), where y = x^3 - 3*x. Iterating the identity produces the product expansion sqrt((x+2)/(x-2)) = product {n = 0..inf} (1 + 2/(a(n) - 1)), with a(0) = x and a(n+1) = a(n)^3 - 3*a(n). The rate of convergence is cubic (Fine).
For similar results to the above see A001566 and A219162.
(End)
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REFERENCES
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E. B. Escott, Rapid method for extracting a square root, Amer. Math. Monthly, 44 (1937), 644-646.
N. J. Fine, Infinite products for k-th roots. Amer. Math. Monthly Vol. 84, No. 8, Oct. 1977
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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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Table of n, a(n) for n=0..4.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart., 11 (1973), 429-437.
Eric Weisstein's World of Mathematics, Pierce Expansion
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FORMULA
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a(n) = 2*F(2*3^n+1)-F(2*3^n) = ceiling(tau^(2*3^n)) where F(k)=A000045(k) is the k-th Fibonacci number and tau is the golden ratio. - Benoit Cloitre, Nov 29 2002
From Peter Bala, Nov 13 2012: (Start)
a(n) = (1/2*(3 + sqrt(5)))^(3^n) + (1/2*(3 - sqrt(5)))^(3^n).
Product {n = 0..inf} (1 + 2/(a(n) - 1)) = sqrt(5).
a(n) = A002814(n+1) + 1.
(End)
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MATHEMATICA
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NestList[#(#^2-3)&, 3, 6] (* Harvey P. Dale, June 09 2011 *)
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PROG
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(PARI) a(n)=2*fibonacci(2*3^n+1)-fibonacci(2*3^n)
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CROSSREFS
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Cf. A006276. A001566, A002814, A112845, A219160, A219161, A219162.
Sequence in context: A157556 A214442 A057133 * A157580 A101293 A189799
Adjacent sequences: A001996 A001997 A001998 * A002000 A002001 A002002
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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