OFFSET
1,1
COMMENTS
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.
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..13
Eric Weisstein's World of Mathematics, Egyptian Fraction.
EXAMPLE
sqrt(3) - 1 = 1/2 + 1/(2*3) + 1/(3*6) + 1/(5*21) + ...
MATHEMATICA
r[k_] := 1/Fibonacci[k+1]; 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[3] - 1; Table[n[x, k], {k, 1, z}]
PROG
(PARI) r(k) = 1/fibonacci(k+1);
f(k, x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x); );
a(k, x=sqrt(3)-1) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 22 2016
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Clark Kimberling, Mar 22 2016
STATUS
approved