%I
%S 1,1,0,4,42,9050,6965359,26721852461,429651752290375,
%T 31194475941824888769,9828395457980805457337560,
%U 13684686862375136981850903785368,83297108604256429529069019958551956425,2226741508593975401942934273354241209226704830,260577257822688861848154672171293101310412373160498171,133631198381015786582155688877301469836628906260462969996612568,299985729493560746632648983353916422875677601725131683097521792924081609
%N Number of meanders filling out an n X n grid, reduced for symmetry.
%C The sequence counts the distinct closed paths that visit every cell of an n X n square lattice at least once, that never cross any edge between adjacent squares more than once, and that do not selfintersect. Paths related by rotation and/or reflection of the square lattice are not considered distinct.
%C Are a(1) and a(2) the only two terms equal to 1? And is a(3) the only term equal to 0?  _Daniel Forgues_, Nov 24 2011
%C The answer is yes: There are several patterns that can straightforwardly be generalized to any grid of any size n>3, e.g., #13 and #6347 of the graphics for a(6) (resp. #24 or #28 of a(5) for odd n).  _M. F. Hasler_, Nov 24 2011
%H Dale Gerdemann, <a href="http://www.youtube.com/watch?v=3qTIrScxWXk">Video illustration for a(5) = 42</a>
%H OEIS Wiki, <a href="/wiki/Number_of_meanders_filling_out_an_nbyn_grid_%28reduced_for_symmetry%29">Number of meanders filling out an nbyn grid (reduced for symmetry)</a>
%H Jon Wild, <a href="/A200000/a200000_1.png">Illustration for a(4) = 4</a>
%H Jon Wild, <a href="/A200000/a200000.png">Illustration for a(5) = 42</a>
%H Jon Wild, <a href="/A200000/a200000_3.png">Illustration for a(6) = 9050</a> [Warning: this is a large file!]
%H Du, Zhao Hui, <a href="/A200000/a200000.cpp.txt">C++ source code for A200000 and A200749</a>
%e a(1) counts the paths that visit the single cell of the 1 X 1 lattice: there is one, the "fat dot".
%e The 4 solutions for n=4, 42 solutions for n=5 and 9050 solutions for n=6 are illustrated in the supporting .png files.
%Y Cf. A200749 (version not reduced for symmetry).
%Y Cf. A200893 (meanders on n X k rectangles instead of squares, reduced for symmetry).
%Y Cf. A201145 (meanders on n X k rectangles, not reduced for symmetry).
%K nonn,nice
%O 1,4
%A _Jon Wild_, Nov 20 2011
%E a(8) and a(10) from _Alex Chernov_, May 28 2012
%E a(9) from _Alex Chernov_, added by _Max Alekseyev_, Jul 21 2013
%E a(11) to a(17) from _Du, Zhao Hui_, Apr 03 2014
