A092297 vs A106512

[Brendan McKay]

The definition of A092297 clearly gives a(1)=0. There is a circular
annulus cut by one radius and you have to colour the single region
so that the radius has a different colour on each side.  This is
impossible so the count is 0.  In graph-theoretic terms it is a
single vertex with a loop on it.

In the case of A106512, your definition is not precise enough to
know what it is supposed to mean for n=1.  I infer from your 
discussion here that you are discarding the loop in that case.

In my opinion, keeping the loop is more natural.  You are counting
sequences  c[0],c[1],...,c[n-1]  where, for each i,
   A.  c[i] \in {1,2,...k}
   B.  c[i] \ne c[(i+1) mod n]
At the moment you are discarding condition B only in the case n=0,
which does not appear natural to me.

Cheers, Brendan.


* Joshua Zucker <joshua.zucker at gmail.com> [050604 02:26]:
> My newly-contributed  sequence, A106512, disagrees with A092297 in its
> first term (view my sequence as square array, and look at column 3).
> A106512: k-colorings of a circle of n nodes.  A092297: the
> 3-colorings.
> 
> I say with one node and 3 colors there are 3 colorings, but A092297
> says there are 0.
> 
> Should A092297 be corrected?
> 
> I can see that the formula (which works for n>1) gives 0 when n = 1,
> but I think that just means the formula only works for n>1.
> 
> There's also a recurrence, which is ALSO broken unless a(1) = 0 -- but
> nonetheless, I think that just means the recurrence has to start with
> a(2) and a(3), and not with a(1) and a(2).
> 
> But maybe my sequence should be corrected instead?
> 
> Thanks,
> --Joshua Zucker