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 A136257 #38 Jul 11 2025 09:06:13 %S A136257 1,20,437,10461,270726,7456194,215666696,6485151199,201183083017, %T A136257 6401210746834,207969967925893,6875935591529309 %N A136257 Number of possible plays on the n-th move in Mirror Chess in which Black's play is always the mirror image of White (White must either mate or play such that Black can mirror the move). %C A136257 By the number of possible plays on the n-th move is meant the total number of legal lines of play for white under the rules of mirror chess at a depth of n moves from the standard initial position. %C A136257 If white cannot play a legal move under the rules of mirror chess then the game is considered to be a draw. %C A136257 Among the 270726 possibilities up to move 4, only 3 correspond to games ending in checkmate, all at move 4: see examples. - _M. F. Hasler_, Dec 08 2021 %H A136257 Jeremy Gardiner, <a href="http://www.woomerang.com/mchess/">Mirror Chess</a> %H A136257 Jeremy Gardiner, <a href="/A136257/a136257.txt">Mirror Chess 3 moves supplied by Francois Labelle</a> %e A136257 From _M. F. Hasler_, Jul 05 2025: (Start) %e A136257 a(1) = 20 because White can make any of the 20 possible starting moves, to each of which Black can reply with the "same" (mirror) move. %e A136257 Then, depending on the starting move, there are between 19 (for 1. a3, f3, f4 and h3) and 30 (for 1. e3) possible moves, for a total of a(2) = 437 moves (for each of which Black can reply with the "same" move). %e A136257 At move 3, there is a total of 48 moves that are legal but not mirror-legal: %e A136257 - After 1.c4: 2.d3 (or d4) 3.Qa4+, 2.d3 3.Qa4+, 2.Qb3 3.Qb5 (or Qxb6), and 2.Qa4 3.Qxa5 (or Qxd7+). %e A136257 - After 1.d4: 2.c4 3.Qa4+, 2.e3 (or e4) 3.Bb5+, and 2.Qd3 3.Qb5+. %e A136257 - Similarly after 1.c3 and 1.d3, and 20 more with 1.e3 or 1.e4. %e A136257 - After 1.f4: 2.e3 3.Qh5+ and 2.e4 3.Qh5+, and after 1.g3: 2.Bh3 3.Bxd7+. %e A136257 - Finally, there are also 9 sequences of only knight moves that end in a check, like 1.Nc3 2.Ne4 3.Nd6+ (or Nf6+). (End) %e A136257 A checkmate cannot occur earlier than at move 4, where we have the following possibilities: 1.d4 d5 2.Qd3 Qd6 3.Qf5 Qf4 4.Qxc8# or 3.Qh3 Qh6 4.Qxc8#, and %e A136257 1.c4 c5 2.Qa4 Qa5 3.Qc6 Qc3 4.Qxc8#, corresponding to the following diagrams: %e A136257 r n Q . k b n r r n Q . k b n r r n Q . k b n r %e A136257 p p p . p p p p p p p . p p p p p p . p p p p p %e A136257 . . . . . . . . . . . . . . . q . . . . . . . . %e A136257 . . . p . . . . . . . p . . . . . . p . . . . . %e A136257 . . . P . q . . . . . P . . . . . . P . . . . . %e A136257 . . . . . . . . . . . . . . . . . . q . . . . . %e A136257 P P P . P P P P P P P . P P P P P P . P P P P P %e A136257 R N B . K B N R R N B . K B N R R N B . K B N R %e A136257 where upper/lowercase letters represent white/black pieces, and dots stand for empty squares. - _M. F. Hasler_, Dec 08 2021 %o A136257 (Python) %o A136257 import chess %o A136257 def A136257(n, B=chess.Board()): %o A136257 if n == 0: return 1 %o A136257 count = 0 %o A136257 for m in B.legal_moves: %o A136257 B.push(m) %o A136257 if not B.is_checkmate(): %o A136257 m.from_square ^= 56 %o A136257 m.to_square ^= 56 # reverse ranks through XOR with 7 %o A136257 if B.is_legal(m): %o A136257 if n == 1: count += 1 %o A136257 else: %o A136257 B.push(m) %o A136257 count += A136257(n - 1, B) %o A136257 B.pop() %o A136257 elif n == 1: count += 1 %o A136257 B.pop() %o A136257 return count # _M. F. Hasler_, Dec 08 2021 %Y A136257 Cf. A048987. %K A136257 nonn,hard,more,fini %O A136257 0,2 %A A136257 _Jeremy Gardiner_, Apr 18 2008 %E A136257 a(2) corrected and a(3) from _Jeremy Gardiner_, Mar 03 2013 %E A136257 a(3) corrected and a(4)-a(11) from _François Labelle_, Apr 12 2015