[seqfan] Re: Rings of regular polygons

Peter Munn techsubs at pearceneptune.co.uk
Fri Mar 31 20:27:44 CEST 2017

Hi Luca,

For some reason, I didn't get to see your post below before sending my
previous one.

Luca Petrone wrote:
> That is (2 n)/(-2 - 2 a + n) must be a positive integer
> for a = 1 to n/2.
> The sequence is the same as A023645, except for N = 6,
> for which A023645 is 2 instead of 1

and clearly the values for 3 and 4 would also differ if the proposed
sequence offset were 3 (or lower) instead of 5.

Which means a sequence in line with Felix's proposal (defined to exclude
n-gon circles where the central hole degenerates to a point) will be new
to OEIS, but match A023645 from a(7) on.  I'll stand back to give Felix or
someone else chance to submit this.

For my part, unless anyone points out a flaw, I will add a comment to
A023645 re circles of edge-sharing, but nonoverlapping n-gons (without any
mention of central holes).  I'm also preparing to add sequences related to
the n-gons that can be defined by the central hole.

Best Regards,

>>----Messaggio originale----
>>Da: "Luca Petrone" <luca.petrone at libero.it>
>>Data: 27/03/2017 7.52
>>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.
>>>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.

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