OFFSET
0,2
COMMENTS
a(n) has the curious property of always being odd but is otherwise quite random. Nevertheless c = lim(n -> infinity) a(n)/n exists, about 2.3926 +/- 0.0004.
FORMULA
The value of lim n -> infinity a(n)/n is log(10)/2/log(phi)=2.3924...
EXAMPLE
phi(6)=1.618033. The continued fraction expansion of phi(6) = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 129}. Hence a(6) = 15.
MATHEMATICA
gr = RealDigits[ N[ GoldenRatio, 250]] [[1]]; f[n_] := Block[ {k = 1}, While[ ContinuedFraction[ FromDigits[ {Take[ gr, n + 1 ], 1} ]] [[k]] == 1, k++ ]; k - 1]; Table[ f[n], {n, 0, 70} ]
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Benoit Cloitre and Boris Gourevitch (boris(AT)314.net), Nov 02 2001
EXTENSIONS
Additional comments from Robert G. Wilson v, Nov 02 2001
STATUS
approved