[seqfan] Re: Big Numbers in the Champernowne Continued Fraction Expansion
Richard Mathar
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
http://list.seqfan.eu/pipermail/seqfan/2010-March/004252.html :
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
sequences.
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.
RJM
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