%I #29 Mar 14 2015 11:38:33
%S 1,1,2,65,5562,893395
%N Number of different patterns using tiles from 1*1 to 1*n with each tile flush to at least one other.
%C Flush means that two tiles have an edge in common.
%C From _Jon Perry_, May 03 2013: (Start)
%C If we require all tiles to be flush to each other, then the sequence is 1, 1, 2, 6, 0, 0, .... with a(n)=0 for n>=4.
%C The 6 patterns for n=3 are:
%C xxx xxx xxx oxxx +xxx xxx
%C oo+ o+ o+ o+ oo oo+
%C o o
%C A proof for a(n)=0 for n>=4 is that these 6 patterns represent all possible 'hinge' patterns for any set of tiles, and by observation no 4th tile is admissible. (end)
%H Giovanni Resta, <a href="/A214366/a214366.png">Illustration of a(3)</a>
%e For n=2 we have:
%e +
%e +oo oo
%e For n=3 some examples are:
%e + o+ o o
%e oo o o o+
%e xxx xxx xxx+ xxx
%e To calculate a(3) we use the 9 basic patterns:
%e o o
%e o o oo oo o
%e xxx xxx xxx xxx oxxx ooxxx
%e 11 6 9 10 11 7
%e + +
%e xxx xxx +xxx
%e 5 2 4
%e and calculate the number of valid positions for the 1*1 tile (top row) and for the 1*2 tile (bottom row).
%K nonn,more
%O 0,3
%A _Jon Perry_, Feb 16 2013
%E a(4) from _Giovanni Resta_, Feb 21 2013
%E a(5) from _Giovanni Resta_, Mar 12 2013