%I #11 Feb 24 2018 11:53:12

%S 2,2,8,99,9153,134325943,17980902816814494,

%T 336913028495678415812394391065577,

%U 70730509948452535771375914216285436007372776802180962851035180747

%N Denominators of r-Egyptian fraction expansion for (1/2)^(1/3), where r(k) = 1/Fibonacci(k+1).

%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="/A270405/b270405.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 (1/2)^(1/3) = 1/2 + 1/(2*2) + 1/(3*8) + 1/(5*99) + ...

%t r[k_] := 1/Fibonacci[k+1]; 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 = (1/2)^(1/3); Table[n[x, k], {k, 1, z}]

%Y Cf. A269993, A000045, A270714.

%K nonn,frac,easy

%O 1,1

%A _Clark Kimberling_, Mar 30 2016