%I #14 Feb 24 2018 10:10:13

%S 2,5,19,409,110469,11663878545,142556979966838173805,

%T 52663280147046053953610628211699561262739,

%U 3562483323353729594120027219074361805521197466091774321103882341800358039125668071

%N Denominators of r-Egyptian fraction expansion for golden ratio - 1, 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="/A270398/b270398.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 = 1/3 + 1/(2*5) + 1/(3*19) + 1/(5*409) + ...

%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 = GoldenRatio - 1; Table[n[x, k], {k, 1, z}]

%o (PARI) r(k) = 1/fibonacci(k+1);

%o f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););

%o a(k, x=(sqrt(5)-1)/2) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Mar 22 2016

%Y Cf. A269993, A000045, A094214.

%K nonn,frac,easy

%O 1,1

%A _Clark Kimberling_, Mar 22 2016