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

1, 1, 2, 65, 5562, 893395

Offset

0,3

Comments

Flush means that two tiles have an edge in common.

From Jon Perry, May 03 2013: (Start)

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.

The 6 patterns for n=3 are:

xxx xxx xxx oxxx +xxx xxx

oo+ o+ o+ o+ oo oo+

o o

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)

Links

Table of n, a(n) for n=0..5.

Giovanni Resta, Illustration of a(3)

Example

For n=2 we have:
     +
+oo  oo
For n=3 some examples are:
+    o+    o    o
oo   o     o    o+
xxx  xxx  xxx+  xxx
To calculate a(3) we use the 9 basic patterns:
o     o
o     o   oo   oo    o
xxx  xxx  xxx   xxx  oxxx  ooxxx
11   6    9    10    11    7
+     +
xxx  xxx  +xxx
5    2    4
and calculate the number of valid positions for the 1*1 tile (top row) and for the 1*2 tile (bottom row).

Crossrefs

Sequence in context: A198665 A185029 A228081 * A220596 A326253 A003821

Adjacent sequences: A214363 A214364 A214365 * A214367 A214368 A214369

Keyword

nonn,more

Author

Jon Perry, Feb 16 2013

Extensions

a(4) from Giovanni Resta, Feb 21 2013

a(5) from Giovanni Resta, Mar 12 2013

Status

approved