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
Eric Weisstein's World of Mathematics, Egyptian Fraction
EXAMPLE
tau - 1 = 1/(1*2) + 1/(2*5) + 1/(6*10) + 1/(24*31) + ...
MATHEMATICA
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 = -1 + GoldenRatio; Table[n[x, k], {k, 1, z}]
PROG
(PARI) r(k) = 1/k!;
f(k, x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x); );
a(k, x=(sqrt(5)-1)/2) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 31 2016
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Clark Kimberling, Mar 30 2016
STATUS
approved