[seqfan] Knight Tour recurrences - same as King Tour

Ron Hardin rhhardin at att.net
Fri Feb 25 23:13:04 CET 2011

```Knight tour counts have the same recurrence as the king tour counts (row 1 of
course is automatic, but for all rows to have the same recurrence still seems
surprising), suggesting now a wild guess that any sort of piece-move will have
that same recurrence; I don't have enough numbers yet to tell.

T(n,k)=Number of n-step knight's tours on a (k+2)X(k+2) board summed over all
starting positions

Table starts
..9...16.....25......36......49......64......81.....100....121....144...169
.16...48.....96.....160.....240.....336.....448.....576....720....880..1056
.16..104....328.....664....1112....1672....2344....3128...4024...5032..6152
.16..208....976....2576....5056....8320...12368...17200..22816..29216.36400
.16..400...2800....9328...21480...39616...63440...92656.127264.167264......
.16..800...8352...34448...91328..186544..322528..498320.712080.............
.16.1280..21664..118480..372384..847520.1584576.2596480....................
.16.2208..57392..405040.1508784.3846192.7777808............................
..0.3184.135184.1290112.5807488............................................
..0.4640.317296.4089632....................................................

Empirical, for all rows: a(n)=3*a(n-1)-3*a(n-2)+a(n-3) for n>3,3,4,6,8,10
respectively for row in 1..6
rhhardin at mindspring.com
rhhardin at att.net (either)

```