cofr keyword usage
David Wilson
davidwwilson at comcast.net
Wed Jun 15 15:42:36 CEST 2005
As with all things OEIS, the "cofr" keyword is not as cut-and-dried as we
would like it to be.
http://www.research.att.com/~njas/sequences/eishelp2.html gives the
definition
cofr: A continued fraction expansion of a number
Therefore, "cofr" should apply only to continued fraction expansions, not to
other sequences related to continued fractions.
If we take the above definition literally, any sequence of positive
integers, except possibly finite sequences ending in 1, is the continued
fraction of some real number, and should therefore be marked "cofr". This
is clearly not what was intended. I'm guessing the proper interpretation is
closer to "standard continued fraction expression for real number
expressible in some other form".
Even by this stricter definition, a lot of continued fractions are not
marked. For example, any periodic sequence of positive integers is the
continued fraction some r = (a+b sqrt(x))/c. We could in theory compute r
for any of these sequences (some sequences, e.g. A000790, with long periods
or large elements, may be practically prohibitive) and add a comment and
"cofr" keyword to them. There are other sequence with closed-form continued
fraction expressions, such as A000027 (natural numbers) for which r is
expressible in terms of elliptic functions. Additionally, there are many
sequences that are not continued fraction expressions, but are marked
"cofr", presumably because they are related to continued fractions. Future
project maybe.
Also, the OEIS does not specify what a continued fraction sequence a_r for a
real number r should look like. I prefer the definition
a_r = (r) if r in Z; (floor(r)) concat a_r(1/(r-floor(r))) otherwise.
I like to index continued fractions starting at a_r(0) = floor(r), your
mileage may vary.
------------------------------------------------------------------------------------------------------
A similar situation occurs with the "base" keyword.
base: Sequence is dependent on base used
Again the definition is a little less than specific. With base-related
sequences, there are usually two types of bases to consider, calculation
bases and expression bases.
Calculation bases are numerical bases used to compute the sequence. For
example, A000040 (the prime numbers) are base-independent (no particular
base representation is required to compute them), so A000040 has no
calculation base. For example, A023416 is the number of zeroes in the
binary representation of n, is computed from the binary representation of n,
so it has 2 as a calculation base. A008560, in which each element is
interpreted in ternary and converted to binary, has calculation bases 2 and
3.
The idea that a particular sequence "uses" or "requires" a particular
calculation base is an ill-defined concept. For example, if A023416 is
defined as the number of zeroes in the binary representation of n, changing
"binary" to "ternary" changes the sequence. On the other hand, using the
alternative definition a(1) = 0; a(2n) = a(n)+1; a(2n+1) = a(n), the
sequence values are independent of whether the calculation is done in binary
of ternary. Any definition that uses a particular base can be recast into
one that does not, so it's often a question of whether the sequence is best
defined in terms of base representations. For example, it is easy to
described the RATS sequence (A004000) in terms of base 10 numerals, whereas
a base-independent definition would be a bear to construct.
Expression bases are bases used to express values in the sequence. The
expression base for A00040 is 10, as is the expression base for A023416.
A008560 appears to have expression base 3. A095425 (10 expressed in various
bases) has several expression bases.
I am thinking that Sloane's original vision of the OEIS was as a database of
integer sequences, and that representational issues would be avoided by
sticking with standard base 10 decimal representation for all elements.
Sequences such as A008560 and A095425, whose elements are not properly
interpreted as base 10 numerals, are actually abuses of the OEIS format.
My working definition of the "base" keyword is "the sequence has a natural
definition that involves at least one calculation base", in other words, at
least one natural definition such that if the base were changed in that
definition, the sequence would change.
Again, I am sure there are lots of infractions to this, or any,
interpretation of "base".
----- Original Message -----
From: "Ralf Stephan" <ralf at ark.in-berlin.de>
To: "seqfan" <seqfan at ext.jussieu.fr>
Sent: Monday, June 13, 2005 8:56 AM
Subject: cofr keyword usage
> Regarding
>
> %I A042599
> %S A042599 1,1,4,9,31,40,2271,2311,9204,20719,71361,92080,5227841,
> %N A042599 Denominators of continued fraction convergents to sqrt(828).
> %K A042599 nonn,cofr,easy
>
> isn't cofr used in the case const. = 1/(a(0)+1/(a(1)+1/.... only?
>
> If so, there might be more mislabeled cofrs.
> If not, then more could be labeled as such.
>
>
> ralf
>
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