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Search: id:A145008
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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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1
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| 5, 41, 3281, 21523361, 926510094425921, 1716841910146256242328924544641, 5895092288869291585760436430706259332839105796137920554548481
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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The recursion x = (n/x + x)/2 converges to a square root of n. The recursion was developed experimentally by analysis of polynomial recursions. See link for the details.
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 (mathar(AT)strw.leidenuniv.nl), Oct 07 2008
The equivalent sequence for n=9 starting from x=1 is 5, 17, 257,.., apparently A000215. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 14 2008
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LINKS
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Cino Hilliard Polynomial roots by recursion
Wikipedia, Newton's method.
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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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PROGRAM
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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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Cf. A059917. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 01 2008]
Sequence in context: A052113 A093433 A065035 this_sequence A025173 A062021 A082145
Adjacent sequences: A145005 A145006 A145007 this_sequence A145009 A145010 A145011
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KEYWORD
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frac,nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)hotmail.com), Sep 28 2008
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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 (mathar(AT)strw.leidenuniv.nl), Oct 14 2008
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