[seqfan] Re: Rings of regular polygons

[Peter Munn]

Hi Luca,

Your equation having "a" in it looks a good way of excluding the solutions
which have the polygons meeting at a point.

If the constraint on "a" is relaxed to allow it to be zero, the number of
solutions to your equation does seem to be the number of divisors of n
that are less than n/2 (which is A023645), which if proven would mean we
reached the same answer by a different analysis routes.  Can you spot any
difference?

If we can be sure the terms for n > 6 match A023645 then it provides an
easy way for Felix or whoever to supply the sequence terms.

Also of interest are the n-gons which define the hole in the middle.  The
sequence of possible n's would appear to form a new sequence for OEIS,
starting 3,4,6,8,9,10, 12,15,16,18,20, 21,24,25,27,28,30, 32,33,36...  and
likewise the number of distinct n-gons that can define such holes.

Regards,

Peter

Luca Petrone wrote:
>
> The number of configurations for rings of n-gons are the number of integer
> solution in k (number of rings) of the equation (a - 1) (1 + 2/n) + 4/n +
> 2/k
> == a, with a (number of sides of the n-gon inside the ring) varying from 1
> to n/2.
>
> Regards,
>
> Luca Petrone
>
>>----Messaggio originale----
>>On Sun, Mar 26, 2017 at 2:15 PM, Peter Munn wrote:
>>>
>>> If you defined it as including all those where neighbouring
>>> polygons shared an edge, that is including 6 triangles around
>>> a point etc., I think the sequence might be A023645, the
>>> number of divisors of n less than n/2.
>>> So, I think your a(13) term should be 1.
>>>
>>> Peter