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A145008
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Reduced numerators of the convergents to 2 = sqrt(4) using the recursion x -> (4/x + x)/2.
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0
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OFFSET
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1,1
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COMMENTS
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The recursion x -> (n/x + x)/2 converges to a square root of n.
These are the numerators of the first order Newton method to solve x^2-4=f(x)=0, starting at x=1 as the initial estimate: x -> x-f(x)/f'(x), where f'(x)=2x is the first derivative. - R. J. Mathar, Oct 07 2008
The equivalent sequence for n=9 starting from x=1 is 5, 17, 257,.., apparently A000215. - R. J. Mathar, Oct 14 2008
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LINKS
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EXAMPLE
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(4/1+1)/2 = 5/2 = 2.5
(4/5/2+5/2)/2 = 41/20 = 2.05
(4/(41/20)+41/20)/2 = 3281/1640 = 2.000609...
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PROG
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(PARI) g(n, p) = x=1; for(j=1, p, x=(n/x+x)/2; print1(numerator(x)", "))
g(4, 8)
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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EXTENSIONS
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Divided the right hand side of formula in the first comment by 2. - R. J. Mathar, Oct 14 2008
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STATUS
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approved
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