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A001601
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a(n) = 2*a(n-1)^2 - 1, if n>1. a(0)=1, a(1)=3.
(Formerly M3042 N1234)
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7
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OFFSET
| 0,2
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COMMENTS
| Reduced numerators of Newton's iteration for sqrt(2). - Eric Weisstein (eric(AT)weisstein.com)
An infinite coprime sequence defined by recursion. - Michael Somos Mar 14 2004
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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. 376.
M. Mendes France and A. J. van der Poorten, From geometry to Euler identities, Theoret. Comput. Sci., 65 (1989), 213-220.
Problem E1093, Amer. Math. Monthly, 61 (1954), 424-425.
J. O. Shallit, Rational numbers with non-terminating, non-periodic modified Engel-type expansions, Fib. Quart., 31 (1993), 37-40.
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
| Dennis Martin, Table of n, a(n) for n = 0..11
J. O. Shallit, Rational numbers with non-terminating, non-periodic modified Engel-type expansions, Fib. Quart., 31 (1993), 37-40.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Pythagoras's Constant
Index entries for sequences related to Engel expansions
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FORMULA
| For n>0: a(n)=a(n-1)^2+2*A051009(n)^2, a(n)^2=2*A051009(n+1)^2+1. - Mario Catalani (mario.catalani(AT)unito.it), May 27 2003
a(n)=sum(Binomial[2^n, 2r]2^r, r=0, .., 2^(n-1)) - Mario Catalani (mario.catalani(AT)unito.it), May 30 2003
Expansion of 1/sqrt(2) as an infinite product: 1/sqrt(2) = prod(k=1, infinity, 1-1/(a(n)+1)). a(1)=3; a(n) = floor(1/(1-1/(sqrt(2)*prod(k=1, n-1, 1-1/(a(k)+1))))) - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Nov 06 2003
A003423(n)=2*a(n+1).
a(n)=(1/2) ((1 + Sqrt[2])^(2^n) + (1 - Sqrt[2])^(2^n)) [From Artur Jasinski (grafix(AT)csl.pl), Oct 10 2008]
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MATHEMATICA
| Table[Simplify[Expand[(1/2) ((1 + Sqrt[2])^(2^n) + (1 - Sqrt[2])^(2^n))]], {n, 0, 7}][From Artur Jasinski (grafix(AT)csl.pl), Oct 10 2008]
Join[{1}, NestList[2#^2-1&, 3, 7]] (* From Harvey P. Dale, Mar 24 2011 *)
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PROG
| (PARI) a(n)=if(n<1, n==0, 2*a(n-1)^2-1)
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CROSSREFS
| Cf. A051009. a(n) = A001333(2^n).
Sequence in context: A098138 A009719 A128300 * A061119 A049985 A126579
Adjacent sequences: A001598 A001599 A001600 * A001602 A001603 A001604
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KEYWORD
| nonn,easy,nice,frac
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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