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Number of 5-step one space leftwards or up, two space rightwards or down asymmetric rook's tours on an n X n board summed over all starting positions.
1

%I #10 Apr 23 2018 08:39:09

%S 0,0,90,416,1208,2400,4092,6208,8766,11752,15166,19008,23278,27976,

%T 33102,38656,44638,51048,57886,65152,72846,80968,89518,98496,107902,

%U 117736,127998,138688,149806,161352,173326,185728,198558,211816,225502,239616

%N Number of 5-step one space leftwards or up, two space rightwards or down asymmetric rook's tours on an n X n board summed over all starting positions.

%C Row 5 of A187296.

%H R. H. Hardin, <a href="/A187300/b187300.txt">Table of n, a(n) for n = 1..50</a>

%F Empirical: a(n) = 214*n^2 - 1080*n + 1152 for n>7.

%F Conjectures from _Colin Barker_, Apr 23 2018: (Start)

%F G.f.: 2*x^3*(45 + 73*x + 115*x^2 - 33*x^3 + 50*x^4 - 38*x^5 + 9*x^6 - 7*x^7) / (1 - x)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>10.

%F (End)

%e Some solutions for 4 X 4:

%e ..0..0..0..0....0..4..3..2....0..0..0..0....0..0..0..0....0..0..0..0

%e ..0..0..2..1....0..0..0..1....5..4..3..0....5..1..0..0....3..0..4..0

%e ..0..0..0..0....0..5..0..0....0..0..2..0....4..0..0..0....2..0..0..0

%e ..0..4..3..5....0..0..0..0....0..0..1..0....3..2..0..0....1..0..5..0

%Y Cf. A187296.

%K nonn

%O 1,3

%A _R. H. Hardin_, Mar 08 2011