[seqfan] Re: Guided browsing of the OEIS based upon personal preferences?
Antti Karttunen
antti.karttunen at gmail.com
Sun Oct 25 19:53:08 CET 2009
(Note: in my previous answer to Rick, some of my own comments
are shown as original quoted comments from Rick. My apologies,
it's just me messing up the reply in the Gmail's composition window).
But back to the classification of the sequences:
in my view, the most fundamental classification of the sequences
is based on their motivating semantics. Whether the foremost
goal was to play with the formula, to find certain kind of numbers or to
count something.
Note that these are not necessarily disjoint categories:
a) sequences with a known "easy" formula, (e.g. recurrence or a generating
function).
Archetypal example: Fibonacci numbers, A000045.
b) sequences which count something (e.g. stable configurations of bricks).
These often have an "easy" formula, at least after somebody finds it, in
which
case they also belong to (a).
Archetypal example: Fibonacci numbers again, A000045.
Sometimes no such formula is found
(like e.g. for http://www.research.att.com/~njas/sequences/A000577 )
thus the cases must be actually _counted_. (Manually or by computer
program.)
Now I wonder, was the original intended usage for "easy" and "hard" keywords
to make this distinction clear? And BTW, A000001 has no "hard" keyword! Why?
c) sequences where membership is determined by some criteria.
Archetypal example: Prime numbers, A000040.
Note that in case a) we can be sure that the sequence is infinite
(provided the formula doesn't go singular/non-integral with some n),
and also with b) it is certain (if the definition of structures to be
counted
does not allow extending it to the infinite sizes, then the rest of terms
are simply zero).
However, with c) we are in general not even sure whether
the sequence is infinite.
Now, of course one can argue that a computer program
for counting some structures or finding integers with
some criteria is also a "formula", and indeed it is,
and in the case of primes it is even a primitive recursive function
(but this hinges on Bertrand's postulate, if I remember
my Math.Log. courses correctly)
But one has to draw a line somewhere?
I wonder whether the concepts from the theory of computation
would actually make this any more practical?
And where do the general "sieve-type" sequences fit, (c) I guess?
Again, random two cents from,
Antti
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