[seqfan] a(1) in A242941

Sun Nov 2 00:31:28 CET 2014

Hello sequence fans,

I thought about A242941 a bit and wondered whether a(1) is correct as
listed. a(1) is the number of uniform tessellations in one-dimensional
space, i.e. the line. A uniform tessellation is a tessellation that is
vertex-transitive and whose facets, tiles or cells are uniform polytopes.
In one dimension, there is only one polytope, namely a line segment bounded
by two points. So, in one dimension, if x-x denotes a line-segment (i.e. a
one dimensional polytope) with x denoting the vertices and - the bounded
'interior', then one possible uniform tessellation is

x-x-x-x-x-x-x-x-x-x-x-x.

If a(1) is correct, then this is the only possible uniform tessellation in
one dimension. However, what about a case like

x-x----x-x----x-x----x-x----x

Would that also count as a uniform tessellation? If so, then a(1) doesn't
seem to be correct, as then there would be infinitely many tessellations.

So is a(1) in A242941 correct?

Best,

Felix