Fwd: Piled polyominos

[Joseph S. Myers]

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

XXX
(same as
X
X
X
by a rotation)

and

X
XX

(same as
 X
XX
by a rotation)

For a(5) = 10,

XXXXX

X
XXXX

 X
XXXX

  X
XXXX

   X
XXXX

XX
XXX

X X
XXX

 XX
XXX

X
X
XXX

 X
 X
XXX

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)

(the cases which can be rotated correspond to partitions, but in the case 
of rectangles there are partitions for both orientations), which I think 
gives

1,1,2,5,10,23,50,108,...

which also doesn't seem to be in the database.

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