Piled polyominos

[franktaw at netscape.net]

Joseph S. Myers jsm at polyomino.org.uk wrote

>On Sat, 15 Oct 2005, franktaw at netscape.net wrote:
>
>>  It starts to get interesting when we allow rotations.  Many piled 
>> polyominos cannot be rotated, but some can.  Allowing rotations but not 
>> reflections, we get a sequence that starts:
>>  
>> 1,1,3,5,11,24,51,110
>
>As I understand this sequence a(3) should be 2, the possibilities being
>...
>If p(n) is the number of partitions of n (A000041) and D(n) is the number 
>of divisors of n that are <= sqrt(n) (A038548) then I think this should be
>
>a(n) = 2^(n-1) - p(n) + D(n)
>
>... which I think gives
>
>1,1,2,5,10,23,50,108,...
>
>which also doesn't seem to be in the database.

This is correct.  I was double-counting the non-square self-dual partitions.
A similar argument gives the sequence eliminating reflection duplicates as:
A005418(n) - A000701(n); A005418 is the total number invariant under reflections (2^(n-2)+2^([n/2]+1), and A000701 is the number of pairs of non-self dual partitions.
>Joseph S. Myers
>jsm at polyomino.org.uk

Franklin T. Adams-Watters16 W. Michigan Ave.Palatine, IL 60067847-776-7645
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