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A269993
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Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r = (1,1/2,1/3,1/4,...)
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99
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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.
Guide to related sequences:
r(k) x denominators
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Using the 12 choices of x shown above (sqrt(1/2) to (1/2)^(1/3), the denominator sequence of the r-Egyptian fraction for x appears for each of the following sequences (r(k)):
r(k) = 1 (see above)
r(k) = 1/k (see above)
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LINKS
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EXAMPLE
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sqrt(1/2) = 1/2 + 1/(2*3) + 1/(3*9) + ...
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MATHEMATICA
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r[k_] := 1/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[1/2]; Table[n[x, k], {k, 1, z}]
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PROG
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(PARI) r(k) = 1/k;
x = sqrt(1/2);
f(x, k) = if(k<1, x, f(x, k - 1) - r(k)/n(x, k));
n(x, k) = ceil(r(k)/f(x, k - 1));
for(k = 1, 10, print1(n(x, k), ", ")) \\ Indranil Ghosh, Mar 27 2017, translated from Mathematica code
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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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