cofr keyword usage

[David Wilson]

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.

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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
>