[seqfan] Re: Is 0001 a 4-digit base 10 number ?

[Marc LeBrun]

Yes, this calls for more careful presentation.  OEIS definitions can entail
unusual interpretations, but they ought to be explicitly surfaced.

At the risk of belaboring this, here's a few guidelines I try to keep in
mind in wordsmithing submissions:

  *  The term "number" can be very slippery and should generally only be
used as a friendlier synonym for "integer" unless qualified in context (as
in "rational number", "real number", "number of", etc).

"Integer" is a precise modern concept that entails philosophical subtleties,
such as identifying integers with measures.  Some ancients would argue, for
example, that zero and one aren't "numbers", because they aren't a QUANTITY
OF things--there are either "NO thing", or "THE thing" or some "NUMBER of
things" (Euclid repeated entire arguments for the special case n=1 and then
all over again for n>1 just because of this).  It is easy to gloss over
these hard-won understandings when using words like "number" loosely.

  *  Most especially, a digit string is not a number--they just often look
like numbers when printed without quotes.  That is, it's sloppy to call "00"
a number or write just 00--it's a "digit string of length two", a "two-digit
string" etc.

  *  The OEIS is an encyclopedia of INTEGER sequences.  Moreover, the
integers giving the actual sequence data are ALWAYS written in decimal.

Consider A007088:
  1, 10, 11, 100, 101, 110, 111,...
It's named "Numbers written in base 2."  Everybody understands what this is
supposed to mean, but it isn't responsibly rigorous in my view.  In fact
A007088 is the sequence of integers that would be written, had the OEIS
instead standardized on Roman numerals, as
  I, X, XI, C, CI, CX, CXI,...
It is NOT the same as 1,2,3,4,5,6,7,... Nor is it the sequence of strings
"1", "10", "11", "100", "101", "110", "111",... Strings can no more can be
an OEIS entry than can values written in hexadecimal, or as factorizations.

Of course it is perfectly OK to map (or count) some class of objects in some
other domain, produce an integer, and then write that integer in decimal as
the value for a(n) in an OEIS sequence.

Confusion can slip in when we relax the rigor with which this provenance is
documented, in order to produce friendlier or more succinct prose.

Hence editorial vigilance is indicated when "number", "written in", "base",
"digit", "leading zeros" and so on appear in descriptions, and careful
rewriting should be considered to help maintain the OEIS's quality.

In this instance, A126364's entry should probably talk explicitly about
"digit strings".  For example, "Number of digit strings of length n such
that adjacent digits differ by at most one."


>="Charles Greathouse" <charles.greathouse at case.edu>

> Also it seems to consider 0 as a 0-digit number -- not the usual
> interpretation, I think.
> 
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
> 
> On Fri, Aug 19, 2011 at 4:39 AM,  <franktaw at netscape.net> wrote:
>> It's not the standard interpretation.
>> 
>> There's nothing wrong with the sequence being defined that way, but either
>> the definition should be modified (e.g., add "with leading zeros allowed"),
>> or a similar comment should be added.
>> 
>> Franklin T. Adams-Watters
>> 
>> -----Original Message-----
>> From: Giovanni Resta <giovanni.resta at iit.cnr.it>
>> 
>> My question arises from the description of A126364
>> ( http://oeis.org/A126364 ) :
>> 
>> Number of base 10 n-digit numbers with adjacent digits differing by one
>> or less.
>> 1, 10, 28, 80, 230, 664,...
>> 
>> It seems clear, from a(2)=28, that the author counts 00 and 01 among
>> the 2-digit base 10 numbers.
>> 
>> It seems to me a little unusual, but I may be wrong.
>> 
>> What's your opinion ?
>> 
>> giovanni
>> 
>> 
>> 
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