two-way infinite sequences
Mitch Harris
maharri at cs.uiuc.edu
Mon Dec 16 04:36:40 CET 2002
On Sat, 14 Dec 2002, Michael Somos wrote:
>I again repeat my call for discussion of two-way infinite
>versus one-way infinite sequences and the implications for
>OEIS entries. Shalom, Michael
(in the context of the 2-way infinite Fibonacci sequence where F(-n) =
(-1)^(n+1) F(n) )
Nobody else bit, so to satisfy the vacuum:
1) Even though the name of the OEIS uses the term "integer", the culture
has encouraged the use of nonnegative integers, anything else is
a special case. Numbers in a particular sequence are always positive; it
is considered an aside that the numbers in a sequence might be more
appropriately signed (the `sign' keyword and -additional- fields of
the signed version).
2) I almost always think of any sequence in the encyclopedia as the
coefficients of a generating function (whether one is known or even exists
doesn't matter). Entries for functions with negative n would be similar to
a Laurent series (a power series with negative integral exponents).
3) Likewise, the OEIS to me is primarily a database of combinatorial
sequences, for which negative parameters, though not disallowed, are
problematic, and really not that common enough to warrant special
changes in the database.
4) But there's an easy way to deal with it already. Why not just have
sequences a_n = some function of (-n)? The `table' feature in a sense
already allows for functions with 2 parameters. A negative parameter (or
even fractional parameters/Puiseux series) can be handled by appropriate
finite combinations of nonnegative sequences.
So all I'm saying is that nothing needs to be done special for two way
infinite sequences, just split into 2 separate sequences, 1 nonnegative, 1
nonpositive.
--
Mitch Harris
Department of Computer Science
University of Illinois at Urbana-Champaign
http://www.uiuc.edu/~maharri
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