Sequence

[franktaw at netscape.net]

I'll try to answer these.  First off, these are both matters of
judgment, not mathematical absolutes.  Thus, the following
represents my opinions, not answers from on high (not that
I'm high -- in any sense -- anyhow).

For offsets, I would apply the following rules, in approximately
this order:

1. If the sequence appears in (mathematical) literature with
an offset specified, use the published offset.  If it appears
with different offsets, use the most common one; if about
equally common, use the following rules to choose between
them.

2. If the description of the sequence mentions the index,
make that description as simple as possible.

3. If you have a formula for the nth term of the sequence,
choose the offset so that that formula is as simple as
possible.

4. If you have a recurrence that mentions the index,
make it as simple as possible.  But note that indices off by
1 are often equally simple; compare

a(n+1) = n*a(n)+a(n-1)
vs.
a(n) = n*a(n-1)+a(n-2)

(This is A001040 and A001053, which both use the
first form).

Note that factorials use the second form, while the
gamma function uses the first -- so this problem is
pervasive.

5. Preserve some symmetry of the sequence, as in your
example.

6. If there are analogous sequences in the database, use
the same offset.

7. Simplify the generating function if you know it.

8. All else failing, use offset 1.

You may still have a choice  of whether to include an
initial 0 -- I would generally include it when appropriate.
E.g., for numbers with a specified property, if 0 has
the property, include it.  An example where I would
not include it is the composite numbers -- by some
definitions, 0 is composite (it has divisors other than
1 and itself); but number-theoretically, it's a special
case, not a composite.

Including the 0 means that if somebody searches for
your sequence including the initial 0, they will find it.
(A search without the 0 will find it either way.)

--------
For two-way infinite sequences, the best is when you can
express a(-n) as some simple function of the positive
sequence (again, as in your example) -- or give some
other simple formula for a(-n).  Otherwise, either forget
it, or submit it as a separate sequence.  If the positive
sequence contains integers while the negative part is
fractions, I would usually forget it.  Generally, it depends
on how interesting the negative part of the sequence
is -- the same criteria as in deciding whether to submit
the sequence in the first place.

Franklin T. Adams-Watters

-----Original Message-----
From: Richard Guy <rkg at cpsc.ucalgary.ca>

Thankyou!  Some other odds & ends:  I've never
understood "offset''.  For this sequence the
"right'' initial values are a(0) = a(1) = 0,
a(2) = a(3) = 1, since then a(-n) = -a(n).
Another mystery that eludes me is what to do
with two-way infinite sequences, though this
one is easy and can be dismissed with the
remark I've just made.
...



On Mar 24 2008, Richard Guy wrote:

>As to divisibility properties, it's easy to see
>that any n-th order recurring sequence will be
>periodic modulo  m  with period bounded by m^n.
>If there are zeroes in the sequence, then  m
>divides the corresponding terms.  Here
>   2|a(n)  just if  n = 0,1,-1 mod 5 = 2^2 + 1
>   3|a(n)    ,,    n = 0,1,-1,6 mod 12 = 3(3+1)
>   5|a(n)    ,,    n = 0,1,-1 mod 13 = (5^2 + 1)/2
>and so on, with corresponding results for higher
>powers of the primes.  For  13  the period is
>13(13 + 1)/2  --  compare 3.  I suspect that 3
>and 13 are divisors of the discriminant of
>     [(2x^2 - x + 2)^2 - 13x^2]/4
>which, so far, I seem to have miscalculated.  R.
>

You are correct, the discriminant is -507 = -3*13^2.

Drew