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%I #12 Feb 23 2018 22:03:30
%S 2,5,19,511,224138,60658204540,203857858414658884506671,
%T 65699957103246706854223474912465037343245580906,
%U 3942313430901049708832516976840058495554562175116278047675351101544028510870033057494673090034
%N Denominators of r-Egyptian fraction expansion for -1 + golden ratio, where r = (1,1/2,1/3,1/4,...)
%C 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.
%C See A269993 for a guide to related sequences.
%H Clark Kimberling, <a href="/A269997/b269997.txt">Table of n, a(n) for n = 1..12</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e tau - 1 = 1/2 + 1/(2*5) + 1/(3*19) + ...
%t r[k_] := 1/k; f[x_, 0] = x; z = 10;
%t n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
%t f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
%t x = GoldenRatio - 1; Table[n[x, k], {k, 1, z}]
%Y Cf. A269993.
%K nonn,frac,easy
%O 1,1
%A _Clark Kimberling_, Mar 15 2016
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