[seqfan] Re: Big Numbers in the Champernowne Continued Fraction Expansion
mathar at strw.leidenuniv.nl
Wed Mar 31 14:48:29 CEST 2010
H^2 celebrated the 10 year anniversary of a sequence, pulling a
cork out of the champagne bottle in
The statistics of handling the offset sequences for cofr sequences
seems to indicate that there is no majority in telling what the offset
for a number x = a(.)+1/(a(.+1)+1/(a(.+2)+... is. Where the associated constant
is x >=1, the sequences usually put floor(x) at a(1). Where x<1, it seems
that an initial term of zero in the sense of x =0 +1/(a(.)+1/(... is often
added explicitly, and this is often put at offset 0. Here, of course,
the inconsistency starts, because this 0 is still at the same
position where the non-zero would appear if x>1.
Whenever one would agree on a standard similar to the one accepted for the
keyword:cons, it is likely that roughly 50 percent of the offsets
would have to change. In addition, the "core" sequences with the cofr keyword
have also a straight definition in terms of some recurrence, which often
starts at a(0) and may contradict such a common offset for all the cofr
So the answer relies on the answer to the question: would one write
down the continued fraction of a number x<1 with an explicit 0 at front?
What is the majority of publications doing in that case?
If the answer is yes, this ought be given index 1 (offset =1) for consistency
with the cases x>1. If the 0 is not written down, the offset is rather =2,
defining x= a(1)+1/(a(2)+1/(a(3)+...) -- in agreement with the convention
for the cons sequences that the indices increase for the subsequent entries.
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