A200000 - OEIS

A200000

Number of meanders filling out an n X n grid, reduced for symmetry.

1, 1, 0, 4, 42, 9050, 6965359, 26721852461, 429651752290375, 31194475941824888769, 9828395457980805457337560, 13684686862375136981850903785368, 83297108604256429529069019958551956425, 2226741508593975401942934273354241209226704830, 260577257822688861848154672171293101310412373160498171, 133631198381015786582155688877301469836628906260462969996612568, 299985729493560746632648983353916422875677601725131683097521792924081609

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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 self-intersect. Paths related by rotation and/or reflection of the square lattice are not considered distinct.

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

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

LINKS

Table of n, a(n) for n=1..17.

Dale Gerdemann, Video illustration for a(5) = 42

OEIS Wiki, Number of meanders filling out an n-by-n grid (reduced for symmetry)

Jon Wild, Illustration for a(4) = 4

Jon Wild, Illustration for a(5) = 42

Jon Wild, Illustration for a(6) = 9050 [Warning: this is a large file!]

Zhao Hui Du, C++ source code for A200000 and A200749

EXAMPLE

a(1) counts the paths that visit the single cell of the 1 X 1 lattice: there is one, the "fat dot".

The 4 solutions for n=4, 42 solutions for n=5 and 9050 solutions for n=6 are illustrated in the supporting .png files.

CROSSREFS