login
A269997
Denominators of r-Egyptian fraction expansion for -1 + golden ratio, where r = (1,1/2,1/3,1/4,...)
2
2, 5, 19, 511, 224138, 60658204540, 203857858414658884506671, 65699957103246706854223474912465037343245580906, 3942313430901049708832516976840058495554562175116278047675351101544028510870033057494673090034
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.
EXAMPLE
tau - 1 = 1/2 + 1/(2*5) + 1/(3*19) + ...
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 = GoldenRatio - 1; Table[n[x, k], {k, 1, z}]
CROSSREFS
Cf. A269993.
Sequence in context: A119550 A119563 A270398 * A270547 A177875 A187602
KEYWORD
nonn,frac,easy
AUTHOR
Clark Kimberling, Mar 15 2016
STATUS
approved