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T(n,k)=Number of (n+2)X(k+2) arrays of permutations of 0..(n+2)*(k+2)-1 with each element moved 0 or 1 knight moves and no more than 1 element left unmoved.
3

%I #5 Nov 10 2015 12:39:24

%S 4,16,16,96,256,96,256,1296,1296,256,5632,28561,67600,28561,5632,9216,

%T 331776,313600,313600,331776,9216,184320,4100625,273566336,106172416,

%U 273566336,4100625,184320,262144,49787136,830361856

%N T(n,k)=Number of (n+2)X(k+2) arrays of permutations of 0..(n+2)*(k+2)-1 with each element moved 0 or 1 knight moves and no more than 1 element left unmoved.

%C Table starts

%C ......4......16........96.......256......5632......9216...184320.262144

%C .....16.....256......1296.....28561....331776...4100625.49787136

%C .....96....1296.....67600....313600.273566336.830361856

%C ....256...28561....313600.106172416

%C ...5632..331776.273566336

%C ...9216.4100625

%C .184320

%F Empirical for column k:

%F k=1: [linear recurrence of order 60]

%F k=2: [order 41]

%e Some solutions for n=3 k=4

%e ..8.14..6..7.17..9....8.12.13.11.17..9....8.14.15.16.17..9....8.12.15.11.17..9

%e .19..3..0..5..2.22...19.18..0..5.14..3....2.18..4..5.23..3...14..3..4..5..2.22

%e ..1.24.10.23.29..4....1..2.25.28.29..4....1..0.25.11.27.28....1..0.25.23.27.28

%e .26.27.28.25.11.15....7..6.24.10.26.27....7..6.24.29.26.10....7..6.24.29.26.10

%e .13.12.18.16.20.21...20.21.22.23.15.16...13.12.22.19.20.21...13.21.18.19.20.16

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Oct 21 2015