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<DIV>Joseph S. Myers <A href="mailto:jsm@polyomino.org.uk">jsm@polyomino.org.uk</A> wrote<BR>
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<DIV class=AOLPlainTextBody id=AOLMsgPart_0_c74acf90-7760-4cd1-9142-fe4078ac5aca><PRE><TT>>On Sat, 15 Oct 2005, <A href="mailto:franktaw%40netscape.net">franktaw@netscape.net</A> 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
>...</TT></PRE><PRE><TT>>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.
</TT></PRE><PRE><TT>This is correct. I was double-counting the non-square self-dual partitions.</TT></PRE><PRE><TT>A similar argument gives the sequence eliminating reflection duplicates as:</TT></PRE><PRE><TT>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.</TT></PRE><PRE><TT>>Joseph S. Myers
><A href="mailto:jsm%40polyomino.org.uk">jsm@polyomino.org.uk</A>
</TT><TT></TT></PRE><PRE><TT>Franklin T. Adams-Watters<BR>16 W. Michigan Ave.<BR>Palatine, IL 60067<BR>847-776-7645</PRE></TT></DIV><!-- end of AOLMsgPart_0_c74acf90-7760-4cd1-9142-fe4078ac5aca --></DIV></DIV>
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