[seqfan] Bent 1X1 Tilings
Ron Hardin
rhhardin at att.net
Sat Dec 21 20:13:12 CET 2013
Taking 2x2 diagonal sum differing from 2x2 antidiagonal sum by S indicating a bent tile with the values giving the corner heights, fill a nxk array with these equally bent tiles (so there are (n+1)x(k+1) points, nXk tiles). Tiles share the two points of their common boundary.
Let the tile corner heights be from 0..V and the diagonals difference S, then problems of the form
-> Numbers of tilings T(n,k,V,S) are the same for
V=a+i, S=a+2i, i=1..infinity probably, for a fixed a>1.
In particular I get the same T(n,k,V,S) counts for (V,S) in four equivalence classes
(3,4) (4,6) (5,8) (6,10)
(4,5) (5,7) (6,9)
(5,6) (6,8) (7,10)
(6,7) (7,9) (8,11)
-> The other odd thing is that all rows and columns n,k have a common recurrence relation.
For the first class, eg., in particular:
/tmp/duh
T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X2 subblock having its diagonal sum differing from its antidiagonal sum by 4 (constant stress 1X1 tilings)
Table starts
...20...40...68..136..236...472...836..1672..3020..6040.11108.22216.41516
...40...68..104..188..304...572...968..1868..3280..6428.11624.22988.42544
...68..104..148..248..380...680..1108..2072..3548..6824.12148.23768.43580
..136..188..248..380..544...908..1400..2492..4096..7628.13208.25340.45664
..236..304..380..544..740..1168..1724..2944..4676..8464.14300.26944.47780
..472..572..680..908.1168..1724..2408..3884..5872.10172.16520.30188.52048
..836..968.1108.1400.1724..2408..3220..4952..7196.12008.18868.33560.56444
.1672.1868.2072.2492.2944..3884..4952..7196..9952.15788.23672.40412.65344
.3020.3280.3548.4096.4676..5872..7196..9952.13220.20080.28988.47776.74756
.6040.6428.6824.7628.8464.10172.12008.15788.20080.28988.39944.62828.93904
Empirical for column k (k=2 recurrence also works for k=1):
k=1: a(n)=2*a(n-1)+3*a(n-2)-6*a(n-3)
k=2: a(n)=3*a(n-1)+3*a(n-2)-15*a(n-3)+4*a(n-4)+18*a(n-5)-12*a(n-6)
k=3: a(n)=3*a(n-1)+3*a(n-2)-15*a(n-3)+4*a(n-4)+18*a(n-5)-12*a(n-6)
k=4: a(n)=3*a(n-1)+3*a(n-2)-15*a(n-3)+4*a(n-4)+18*a(n-5)-12*a(n-6)
k=5: a(n)=3*a(n-1)+3*a(n-2)-15*a(n-3)+4*a(n-4)+18*a(n-5)-12*a(n-6)
k=6: a(n)=3*a(n-1)+3*a(n-2)-15*a(n-3)+4*a(n-4)+18*a(n-5)-12*a(n-6)
k=7: a(n)=3*a(n-1)+3*a(n-2)-15*a(n-3)+4*a(n-4)+18*a(n-5)-12*a(n-6)
All.solutions.for.n=k=1..
..0..2....2..0....0..2....2..0....0..3....2..1....0..1....0..3....0..3....2..3..
..2..0....0..2....3..1....1..3....2..1....0..3....3..0....1..0....3..2....3..0..
..
..1..0....3..0....3..0....3..2....1..2....3..0....1..3....3..1....1..3....3..1..
..0..3....0..1....2..3....0..3....3..0....1..2....2..0....0..2....3..1....1..3..
..
/tmp/duq
T(n,k)=Number of (n+1)X(k+1) 0..4 arrays with every 2X2 subblock having its diagonal sum differing from its antidiagonal sum by 6 (constant stress 1X1 tilings)
Table starts
...20...40...68..136..236..472..836.1672.3020.6040
...40...68..104..188..304..572..968.1868.3280.....
...68..104..148..248..380..680.1108.2072..........
..136..188..248..380..544..908.1400...............
..236..304..380..544..740.1168....................
..472..572..680..908.1168.........................
..836..968.1108.1400..............................
.1672.1868.2072...................................
.3020.3280........................................
.6040.............................................
/tmp/dux
T(n,k)=Number of (n+1)X(k+1) 0..5 arrays with every 2X2 subblock having its diagonal sum differing from its antidiagonal sum by 8 (constant stress 1X1 tilings)
Table starts
...20...40...68..136..236..472..836.1672.3020.6040
...40...68..104..188..304..572..968.1868.3280.....
...68..104..148..248..380..680.1108.2072..........
..136..188..248..380..544..908.1400...............
..236..304..380..544..740.1168....................
..472..572..680..908.1168.........................
..836..968.1108.1400..............................
.1672.1868.2072...................................
.3020.3280........................................
.6040.............................................
/tmp/dvd
T(n,k)=Number of (n+1)X(k+1) 0..6 arrays with every 2X2 subblock having its diagonal sum differing from its antidiagonal sum by 10 (constant stress 1X1 tilings)
Table starts
...20...40...68..136..236..472..836.1672.3020.6040
...40...68..104..188..304..572..968.1868.3280.....
...68..104..148..248..380..680.1108.2072..........
..136..188..248..380..544..908.1400...............
..236..304..380..544..740.1168....................
..472..572..680..908.1168.........................
..836..968.1108.1400..............................
.1672.1868.2072...................................
.3020.3280........................................
.6040.............................................
rhhardin at mindspring.com
rhhardin at att.net (either)
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