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Number of 8-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 14:27:04

%S 0,0,72,2028,12778,37124,81362,145100,231552,338044,465734,613212,

%T 780752,968044,1175118,1401948,1648534,1914876,2200974,2506828,

%U 2832438,3177804,3542926,3927804,4332438,4756828,5200974,5664876,6148534,6651948,7175118

%N Number of 8-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 8 of A187296.

%H R. H. Hardin, <a href="/A187303/b187303.txt">Table of n, a(n) for n = 1..50</a>

%F Empirical: a(n) = 9878*n^2 - 79388*n + 143388 for n>13.

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

%F G.f.: 2*x^3*(36 + 906*x + 3455*x^2 + 2401*x^3 + 3148*x^4 - 196*x^5 + 1607*x^6 - 1337*x^7 + 579*x^8 - 705*x^9 + 137*x^10 - 155*x^11 + 15*x^12 - 13*x^13) / (1 - x)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>16.

%F (End)

%e Some solutions for 4 X 4:

%e ..0..1..0..0....1..6..5..0....0..6..5..7....6..0..7..0....8..7..6..2

%e ..0..6..5..7....0..0..4..0....0..0..4..0....5..1..0..0....0..0..5..1

%e ..3..2..4..0....2..7..3..8....2..1..3..8....4..0..8..0....0..0..4..3

%e ..0..0..0..8....0..0..0..0....0..0..0..0....3..2..0..0....0..0..0..0

%Y Cf. A187296.

%K nonn

%O 1,3

%A _R. H. Hardin_, Mar 08 2011