[seqfan] Tiling and Diagonal Sums

[Ron Hardin]

An equivalence that I don't exactly understand, but must have to do with joining 
possibilities

T(n,k)=Half the number of (n+1)X(k+1) 0..2 arrays with every 2X2 subblock 
diagonal sum differing from its antidiagonal sum by more than 2

T(n,k) apparently is also the number of ways to tile a (n+2)X(k+2) rectangle 
with 1X1 and 2X2 tiles

Table starts
....5.....11......21........43.........85.........171...........341
...11.....35......93.......269........747........2115..........5933
...21.....93.....314......1213.......4375.......16334.........59925
...43....269....1213......6427......31387......159651........795611
...85....747....4375.....31387.....202841.....1382259.......9167119
..171...2115...16334....159651....1382259....12727570.....113555791
..341...5933...59925....795611....9167119...113555791....1355115601
..683..16717..221799...4005785...61643709..1029574631...16484061769
.1365..47003..817280..20064827..411595537..9258357134..198549329897
.2731.132291.3018301.100764343.2758179839.83605623809.2403674442213

Diagonal is A063443(n+2)
Column 1 is A001045(n+3)
Column 2 is A054854(n+2)
Column 3 is A054855(n+2)
Column 4 is A063650(n+2)
Column 5 is A063651(n+2)
Column 6 is A063652(n+2)
Column 7 is A063653(n+2)
Column 8 is A063654(n+2)

Some solutions for 6X6  
..0..2..0..2..0..2....0..1..0..2..1..2....0..2..0..2..0..2....0..1..0..2..0..1..
..2..0..2..0..2..1....2..0..2..0..2..0....2..0..1..0..1..0....2..0..2..0..2..0..
..0..2..0..2..0..2....1..2..1..2..0..2....0..2..0..2..0..2....0..2..0..2..0..2..
..2..0..2..0..2..1....2..0..2..0..1..0....1..0..2..0..2..0....1..0..2..0..2..0..
..0..2..0..2..0..2....0..2..0..2..0..2....0..2..0..2..0..2....0..2..1..2..1..2..
..1..0..1..0..1..0....2..1..2..1..2..0....2..1..2..1..2..1....2..0..2..0..2..0..


 rhhardin at mindspring.com
rhhardin at att.net (either)