This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383153 #58 Apr 25 2025 14:31:07 %S A383153 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, %T A383153 62,1,1,170,6004,25111,25111,6004,170,1,1,469,46488,455219,882130, %U A383153 455219,46488,469,1,1,1297,367880,9103712,36979379,36979379,9103712,367880,1297,1 %N A383153 Square array read by antidiagonals: A(m,n) is the number of 2m-by-2n fers-wazir tours. %C A383153 The simplest fairy chess pieces, going back to 9th-century Persia, are the fers -- a (1,1) leaper -- and the wazir -- a (1,0) leaper. (A king combines the moves of a fers and a wazir.) A fers-wazir tour visits every cell of a board exactly once, making fers and wazir moves alternately, and returns to the starting cell. %C A383153 Such tours exist only when the number of rows is even and the number of columns is even. %C A383153 For fixed m, these tours can be enumerated with the transfer-matrix method, so the numbers A(m,n) satisfy a linear recurrence. %D A383153 D. E. Knuth, Hamiltonian paths and cycles, Section 7.2.2.4 of The Art of Computer Programming (to appear). %H A383153 D. E. Knuth, <a href="/A383153/b383153.txt">Table of n, a(n) for n = 1..66</a> %H A383153 George Jelliss, <a href="https://www.mayhematics.com/t/1n.htm">Introducing Knight's Tours</a>, has a 9th century example of a fers-knight tour due to As-Suli. %F A383153 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 %e A383153 Array begins: (example extended by _Filip Stappers_, Apr 21 2025) %e A383153 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... %e A383153 1, 2, 4, 9, 23, 62, 170, 469, 1297, 3590, 9940, 27525, ... %e A383153 1, 4, 22, 124, 818, 6004, 46448, 367880, ... %e A383153 1, 9, 124, 1620, 25111, 455219, 9103712, ... %e A383153 1, 23, 818, 25111, 882130, 36979379, ... %e A383153 1, 62, 6004, ... %e A383153 1, 170, ... %e A383153 1, ... %e A383153 ... %e A383153 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): %e A383153 . %e A383153 a-x e-d i-h a w-v p-q s a w-v s-r p a w-v d-e g %e A383153 X X X |X X X| |X X X| |X X X| %e A383153 w b-c f-g j x b o u-t r x b t-u o q x b-c u h f %e A383153 | | | | | | | | %e A383153 v s-r o-n k e c n h-i k e c i-h n l q o-n t i k %e A383153 X X X |X X X| |X X X| |X X X| %e A383153 t-u p-q l-m d f-g m-l j d f-g j-k m p r-s m-l j %Y A383153 Cf. A383154 (the diagonal A(n,n)). %Y A383153 Cf. A339190 (the analog for king tours). %K A383153 nonn,tabl %O A383153 1,1 %A A383153 _Don Knuth_, Apr 18 2025