[seqfan] King Tour recurrence

[Ron Hardin]

Playing around with all the chess pieces, and working out the king,

T(n,k)=Number of n-step king's tours on a kXk board summed over all starting 
positions

Table starts
.1..4....9.....16.......25.......36........49.......64.......81......100
.0.12...40.....84......144......220.......312......420......544......684
.0.24..160....408......768.....1240......1824.....2520.....3328.....4248
.0.24..496...1764.....3768.....6508......9984....14196....19144....24828
.0..0.1208...6712....17280....32520.....52432....77016...106272...140200
.0..0.2240..22672....74072...156484....268048...408764...578632...777652
.0..0.2984..68272...296360...722384...1335984..2129440..3102752..4255920
.0..0.2384.183472..1110000..3193800...6481216.10899404.16418600.23038804
.0..0..784.436984..3908376.13530576..30543072.54738536.85743256.........
.0..0....0.905776.12956800.55056168.139775784...........................

Empirical, for all rows: a(n)=3*a(n-1)-3*a(n-2)+a(n-3) for n>3,3,3,5,6,7,8,9 
respectively for row=1..8

all rows have the same recurrence (I think), for large enough index.

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