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A051009
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Reduced denominators of Newton's iteration for sqrt(2).
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3
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OFFSET
| 1,2
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COMMENTS
| For n>=4, A051009(n) = A098890(n-2) - A098890(n-3) - Kieren MacMillan (kieren(AT)alumni.rice.edu), Dec 19 2007
(2^n)-th Pell numbers [From Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Dec 04 2008]
For n>1, egyptian fraction expansion of 2-sqrt(2), i.e. 2-sqrt(2) = 1/2 + 1/12 + 1/408 + 1/470832 + ...
Simon Plouffe, Feb 22. 2011.
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LINKS
| 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
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FORMULA
| a(n) = 2*a(n-1)*A001601(n-1) - Joe Keane (jgk(AT)jgk.org), May 31 2002
sqrt(2) = 1 + 1/2 - sum_{n=3, infinity} (1/a(n)). - Donald S. McDonald (don.mcdonald(AT)paradise.net.nz), Jan 21 2003
For n>1: a(n)=2a(n-1)*sqrt(2a(n-1)^2+1). - Mario Catalani (mario.catalani(AT)unito.it), May 27 2003
For n>0: a(n)=sum(binomial[2^n, 2r+1]2^r, r=0, .., 2^(n-1)-1) - Mario Catalani (mario.catalani(AT)unito.it), May 30 2003
a(n)=(1/(2 Sqrt[2])) ((1 + Sqrt[2])^(2^n) - (1 - Sqrt[2])^(2^n))]], {n, 0, 7}] [From Artur Jasinski (grafix(AT)csl.pl), Oct 10 2008]
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MATHEMATICA
| Table[Simplify[Expand[(1/(2 Sqrt[2])) ((1 + Sqrt[2])^(2^n) - (1 - Sqrt[2])^(2^n))]], {n, 0, 7}] [From Artur Jasinski (grafix(AT)csl.pl), Oct 10 2008]
Do[Print[Fibonacci[2^n, 2]], {n, 0, 10}] [From Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Dec 04 2008]
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CROSSREFS
| Cf. A001601.
a(n) = A000129(2^n).
Cf. A098890.
A000129 [From Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Dec 04 2008]
Sequence in context: A009706 A012551 A156509 * A060942 A072446 A015181
Adjacent sequences: A051006 A051007 A051008 * A051010 A051011 A051012
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KEYWORD
| nonn,frac
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com)
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