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A129187
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Decimal expansion of arcsinh(1/3).
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3
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3, 2, 7, 4, 5, 0, 1, 5, 0, 2, 3, 7, 2, 5, 8, 4, 4, 3, 3, 2, 2, 5, 3, 5, 2, 5, 9, 9, 8, 8, 2, 5, 8, 1, 2, 7, 7, 0, 0, 5, 2, 4, 5, 2, 8, 9, 9, 0, 7, 6, 7, 4, 5, 1, 2, 7, 5, 6, 2, 9, 5, 1, 5, 4, 2, 7, 1, 7, 6, 5, 6, 2, 9, 4, 9, 3, 2, 7, 2, 1, 4, 1, 1, 9, 8, 2, 4, 7, 7, 3, 0, 6, 3, 2, 3, 1, 9, 5, 5
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OFFSET
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0,1
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COMMENTS
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Archimedes's-like scheme: set p(0) = 1/sqrt(10), q(0) = 1/3; p(n+1) = 2*p(n)*q(n)/(p(n)+q(n)) (arithmetic mean of reciprocals, i.e., 1/p(n+1) = (1/p(n) + 1/q(n))/2), q(n+1) = sqrt(p(n+1)*q(n)) (geometric mean, i.e., log(q(n+1)) = (log(p(n+1)) + log(q(n)))/2), for n >= 0. The error of p(n) and q(n) decreases by a factor of approximately 4 each iteration, i.e., approximately 2 bits are gained by each iteration. Set r(n) = (2*q(n) + p(n))/3, the error decreases by a factor of approximately 16 for each iteration, i.e., approximately 4 bits are gained by each iteration. For a similar scheme see also A244644. - A.H.M. Smeets, Jul 12 2018
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LINKS
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FORMULA
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EXAMPLE
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0.32745015023725844332253525998825812770052452899076745127562...
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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