A383153 Square array read by antidiagonals: A(m,n) is the number of 2m-by-2n fers-wazir tours.
2, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 9, 22, 9, 1, 1, 23, 124, 124, 23, 1, 1, 62, 818, 1620, 818, 62, 1, 1, 170, 6004, 25111, 25111, 6004, 170, 1, 1, 469, 46488, 455219, 882130, 455219, 46488, 469, 1, 1, 1297, 367880, 9103712, 36979379, 36979379, 9103712, 367880, 1297, 1
Offset: 1
Examples
Array begins: (example extended by _Filip Stappers_, Apr 21 2025) 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 2, 4, 9, 23, 62, 170, 469, 1297, 3590, 9940, 27525, ... 1, 4, 22, 124, 818, 6004, 46448, 367880, ... 1, 9, 124, 1620, 25111, 455219, 9103712, ... 1, 23, 818, 25111, 882130, 36979379, ... 1, 62, 6004, ... 1, 170, ... 1, ... ... For m = 2 and n = 3, the A(2,3) = 4 solutions are the following 4-by-6 tours (a to b to ... to x): . a-x e-d i-h a w-v p-q s a w-v s-r p a w-v d-e g X X X |X X X| |X X X| |X X X| w b-c f-g j x b o u-t r x b t-u o q x b-c u h f | | | | | | | | v s-r o-n k e c n h-i k e c i-h n l q o-n t i k X X X |X X X| |X X X| |X X X| t-u p-q l-m d f-g m-l j d f-g j-k m p r-s m-l j
References
- D. E. Knuth, Hamiltonian paths and cycles, Section 7.2.2.4 of The Art of Computer Programming (to appear).
Links
- D. E. Knuth, Table of n, a(n) for n = 1..66
- George Jelliss, Introducing Knight's Tours, has a 9th century example of a fers-knight tour due to As-Suli.
Formula
G.f. of column 2: z*(1 - 2*z - z^3)/((1 - z)*(1 - 3*z + z^2 - z^3)). - Filip Stappers, Apr 21 2025
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