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A001566
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a(0) = 3; thereafter, a(n) = a(n-1)^2 - 2.
(Formerly M2705 N1084)
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28
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OFFSET
| 0,1
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COMMENTS
| Expansion of 1/phi: 1/phi = (1-1/3)*(1-1/((3-1)*7))*(1-1/(((3-1)*7-1)*47))*(1-1/((((3-1)*7-1)*47-1)*2207))... (phi being the golden ration (1+sqrt(5))/2) - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Nov 06 2003
An infinite coprime sequence defined by recursion. - Michael Somos Mar 14 2004
Starting with 7, the terms end with 7,47,07,47,07,..., of the form 8a+7 where a = 0,1,55,121771,... Conjecture: Every a is squarefree, every other a is divisible by 55, the a's are a subset of A046194, the heptagonal triangular numbers (the first,2nd,3rd,6th,11th,?... terms) . - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 08 2004
Also the reduced numerator of the convergents to sqrt(5) using Newton's recursion x = (5/x+x)/2. [Cino Hilliard (hillcino368(AT)hotmail.com), Sep 28 2008]
The subsequence of primes begins a(n) for n = 0, 1, 2, 3. [Jonathan Vos Post, Feb 26, 2011].
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REFERENCES
| L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 397.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 223.
E. Lucas, Nouveaux theoremes d'arithmetique superieure, Comptes Rend., 83 (1876), 1286-1288.
M. Mendes France and A. J. van der Poorten, From geometry to Euler identities, Theoret. Comput. Sci., 65 (1989), 213-220.
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
| Vincenzo Librandi, Table of n, a(n) for n = 0..12
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart., 11 (1973), 429-437.
Index entries for sequences of form a(n+1)=a(n)^2 + ...
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FORMULA
| a(n) = Fibonacci(2^(n+2))/Fibonacci(2^(n+1)) = A058635(n+2)/A058635(n+1). - Len Smiley, May 08 2000, and Artur Jasinski, Oct 05 2008
a(n) = ceiling(c^(2^n)) where c = (3+sqrt(5))/2 = tau^2 is the largest root of x^2-3*x+1=0. - Benoit Cloitre, Dec 03, 2002
a(n) = round(G^(2^n)) where G is the golden ratio (A001622). - Artur Jasinski, Sep 22 2008
a(n) = (G^(2^(n+1))-(1-G)^(2^(n+1)))/((G^(2^n))-(1-G)^(2^n)) = G^(2^n)+(1-G)^(2^n) = G^(2^n)+(-G)^(-2^n) where G is the golden ratio. - Artur Jasinski, Oct 05 2008
a(n) = 2*cosh(2^n*arccosh(sqrt(5)/2). - Artur Jasinski, Oct 09 2008
a(n) = Fibonacci(2^(n+1)-1)+Fibonacci(2^(n+1)+1). (3-sqrt(5))/2 = 1/3 + 1/(3*7) + 1/(3*7*47) + 1/(3*7*47*2207) + ... (E. Lucas) - Philippe DELEHAM, Apr 21 2009
a(n)*(a(n+1)-1)/2 = A023039(2^n). - M. F. Hasler, Sep 27 2009
For n>=1, a(n) = 2+prod{i=0..n-1}(a(i)+2). - Vladimir Shevelev, Nov 28 2010]
a(n) = 2*T(2^n,3/2) where T(n,x) is the Chebyshev polynomial of the first kind. - Leonid Bedratyuk, Mar 17 2011
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EXAMPLE
| Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Sep 28 2008: (Start)
Init x=1.
x = (5/1+1)/2 = 3/1
x = (5/3+3)/2 = 7/3
x = (5/7/3+7/3)/2 = 47/21
x = (5/47/21+47/21)/2 = 2207/987
(2207/987)^2 = 5.000004106... (End)
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MAPLE
| a:= n-> simplify(2*ChebyshevT(2^n, 3/2), 'ChebyshevT'):
seq (a(n), n=0..8);
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MATHEMATICA
| c = N[GoldenRatio, 1000]; Table[Round[c^(2^n)], {n, 1, 10}] [From Artur Jasinski (grafix(AT)csl.pl), Sep 22 2008]
c = (1 + Sqrt[5])/2; Table[Expand[c^(2^n) + (-c + 1)^(2^n)], {n, 1, 8}] [From Artur Jasinski (grafix(AT)csl.pl), Oct 05 2008]
G = (1 + Sqrt[5])/2; Table[Expand[(G^(2^(n + 1)) - (1 - G)^(2^(n + 1)))/Sqrt[5]]/Expand[((G^(2^n)) - (1 - G)^(2^n))/Sqrt[5]], {n, 1, 10}] [From Artur Jasinski (grafix(AT)csl.pl), Oct 05 2008]
Table[2*Cosh[2^n*ArcCosh[Sqrt[5]/2], {n, 1, 30}] [From Artur Jasinski (grafix(AT)csl.pl), Oct 09 2008]
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PROG
| (PARI) a(n)=if(n<1, 3*(n==0), a(n-1)^2-2)
Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Sep 28 2008: (Start)
(PARI) g(n, p) = x=1; for(j=1, p, x=(n/x+x)/2; print1(numerator(x)", "))
g(5, 8) (End)
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CROSSREFS
| Lucas numbers (A000032) with subscripts that are powers of 2 greater than 1 (Herb Wilf). Cf. A000045.
Cf. A003010 (starting with 4), A003423 (starting with 6), A003487 (starting with 5).
Cf. A058635. [From Artur Jasinski (grafix(AT)csl.pl), Oct 05 2008]
Sequence in context: A020754 A052381 A031440 * A173771 A019039 A077559
Adjacent sequences: A001563 A001564 A001565 * A001567 A001568 A001569
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KEYWORD
| easy,nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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