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A187850 T(n,k) is the number of n-step king-knight's tours (piece capable of both kinds of moves) on a k X k board summed over all starting positions.

%I A187850 #12 Nov 29 2024 17:35:43
%S A187850 1,4,0,9,12,0,16,56,24,0,25,132,304,24,0,36,240,1056,1400,0,0,49,380,
%T A187850 2312,7620,5328,0,0,64,552,4048,20952,49776,16032,0,0,81,756,6264,
%U A187850 41652,177040,292776,35328,0,0,100,992,8960,69456,408048,1400168,1533064,49536,0,0
%N A187850 T(n,k) is the number of n-step king-knight's tours (piece capable of both kinds of moves) on a k X k board summed over all starting positions.
%H A187850 R. H. Hardin, <a href="/A187850/b187850.txt">Table of n, a(n) for n = 1..99</a>
%F A187850 Empirical: T(1,k) = k^2.
%F A187850 Empirical: T(2,k) = 16*k^2 - 36*k + 20.
%F A187850 Empirical: T(3,k) = 240*k^2 - 904*k + 832 for k>3.
%F A187850 Empirical: T(4,k) = 3504*k^2 - 17748*k + 21996 for k>5.
%F A187850 Empirical: T(5,k) = 50128*k^2 - 312688*k + 476944 for k>7.
%F A187850 Empirical: T(6,k) = 706880*k^2 - 5180252*k + 9274644 for k>9.
%F A187850 Empirical: T(7,k) = 9862808*k^2 - 82444808*k + 168212080 for k>11.
%F A187850 Empirical: T(8,k) = 136526552*k^2 - 1275583564*k + 2906368876 for k>13.
%e A187850 Table starts:
%e A187850 .1..4.....9.......16........25........36........49........64.......81.....100
%e A187850 .0.12....56......132.......240.......380.......552.......756......992....1260
%e A187850 .0.24...304.....1056......2312......4048......6264......8960....12136...15792
%e A187850 .0.24..1400.....7620.....20952.....41652.....69456....104268...146088..194916
%e A187850 .0..0..5328....49776....177040....408048....744696...1183632..1723120.2362864
%e A187850 .0..0.16032...292776...1400168...3807828...7700944..13082348.19910456
%e A187850 .0..0.35328..1533064..10353632..33908456..76860784.140714528
%e A187850 .0..0.49536..7067600..71450504.288493336.741624088
%e A187850 .0..0.32256.28260592.458862208
%e A187850 .0..0.....0.96217616
%e A187850 Some n=4 solutions for 4 X 4:
%e A187850 ..1..2..0..0....0..1..0..0....1..0..0..0....0..0..0..0....0..0..0..4
%e A187850 ..0..0..3..0....2..0..0..0....0..2..0..0....0..0..0..0....0..1..0..3
%e A187850 ..0..0..0..0....0..3..0..0....0..3..0..0....0..2..0..0....0..0..2..0
%e A187850 ..0..0..0..4....0..0..0..4....0..4..0..0....0..1..3..4....0..0..0..0
%Y A187850 Row 2 is A104188(n-1).
%K A187850 nonn,tabl
%O A187850 1,2
%A A187850 _R. H. Hardin_, Mar 14 2011