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A270348
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Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r = (1,1/2,1/4,1/8,...)
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1
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2, 7, 43, 1161, 796510, 1101781866330, 648667164391834988511313, 521313118065995695198529265268104396429334449023, 177042477384698216444912803612486097958997328262217304760270340328784709181787835108737458616981
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OFFSET
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1,1
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COMMENTS
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Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ... , the r-Egyptian fraction for x.
See A269993 for a guide to related sequences.
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LINKS
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EXAMPLE
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sqrt(1/3) = 1/2 + 1/(2*7) + 1/(4*43) + ...
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MATHEMATICA
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r[k_] := 2/2^k; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt[1/3]; Table[n[x, k], {k, 1, z}]
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PROG
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(PARI) r(k) = 2/2^k;
f(k, x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x); );
a(k, x=sqrt(1/3)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 18 2016
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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