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A214366 Number of different patterns using tiles from 1*1 to 1*n with each tile flush to at least one other. 2

%I

%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

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Last modified October 26 02:14 EDT 2021. Contains 348256 sequences. (Running on oeis4.)