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A187189 T(n,k) = Number of n-turn rook's tours on a k X k board summed over all starting positions.

%I A187189 #11 Sep 03 2022 20:28:19
%S A187189 1,4,0,9,8,0,16,36,8,0,25,96,108,8,0,36,200,480,288,0,0,49,360,1400,
%T A187189 2208,720,0,0,64,588,3240,9200,9792,1440,0,0,81,896,6468,27720,58800,
%U A187189 40896,2304,0,0,100,1296,11648,68208,231840,364800,156672,2664,0,0,121,1800
%N A187189 T(n,k) = Number of n-turn rook's tours on a k X k board summed over all starting positions.
%H A187189 R. H. Hardin, <a href="/A187189/b187189.txt">Table of n, a(n) for n = 1..100</a>
%F A187189 Empirical: T(1,k) = k^2.
%F A187189 Empirical: T(2,k) = 2*k^3 - 2*k^2.
%F A187189 Empirical: T(3,k) = 4*k^4 - 10*k^3 + 6*k^2.
%F A187189 Empirical: T(4,k) = 8*k^5 - 34*k^4 + 48*k^3 - 22*k^2.
%F A187189 Empirical: T(5,k) = 16*k^6 - 98*k^5 + 228*k^4 - 238*k^3 + 92*k^2.
%F A187189 Empirical: T(6,k) = 32*k^7 - 258*k^6 + 846*k^5 - 1426*k^4 + 1234*k^3 - 428*k^2.
%F A187189 Empirical: T(7,k) = 64*n^8 - 642*n^7 + 2718*n^6 - 6346*n^5 + 8770*n^4 - 6788*n^3 + 2224*n^2.
%F A187189 Empirical: T(8,k) = 128*k^9 - 1538*k^8 + 7956*k^7 - 23556*k^6 + 44586*k^5 - 55218*k^4 + 40894*k^3 - 13252*k^2.
%e A187189 Table starts
%e A187189 .1.4....9......16.......25........36........49........64.......81.....100
%e A187189 .0.8...36......96......200.......360.......588.......896.....1296....1800
%e A187189 .0.8..108.....480.....1400......3240......6468.....11648....19440...30600
%e A187189 .0.8..288....2208.....9200.....27720.....68208....146048...282528..505800
%e A187189 .0.0..720....9792....58800....231840....705600...1800960..4046112.8251200
%e A187189 .0.0.1440...40896...364800...1902240...7197120..21960960.57407616
%e A187189 .0.0.2304..156672..2169600..15220800..72182880.264520704
%e A187189 .0.0.2664..551232.12319200.118332000.709921800
%e A187189 .0.0.1512.1760256.66746400.893246400
%e A187189 .0.0....0.5013504
%e A187189 Some n=3 solutions for 3 X 3
%e A187189 ..0..0..1....0..0..3....3..2..0....1..0..0....0..0..1....0..0..0....0..0..3
%e A187189 ..0..0..0....0..0..2....0..1..0....0..0..0....0..3..2....0..0..0....0..0..1
%e A187189 ..0..3..2....0..0..1....0..0..0....2..0..3....0..0..0....1..3..2....0..0..2
%Y A187189 Row 2 is A035006.
%K A187189 nonn,tabl
%O A187189 1,2
%A A187189 _R. H. Hardin_, Mar 06 2011