

A201145


Triangle read by rows: number of meanders filling out an n X k grid, unreduced for symmetry.


3



1, 0, 1, 0, 1, 0, 0, 1, 2, 11, 0, 1, 2, 42, 320, 0, 1, 6, 199, 3278, 71648, 0, 1, 10, 858, 29904, 1369736, 55717584, 0, 1, 22, 3881, 285124, 27876028, 2372510658, 213773992667, 0, 1, 42, 17156, 2671052, 549405072, 98927211122, 18677872557034, 3437213982024260
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OFFSET

1,9


COMMENTS

This sequence counts the closed paths that visit every cell of an n X k rectangular lattice at least once, never cross any edge between adjacent squares more than once, and do not selfintersect. Paths related by rotation and/or reflection of the square lattice are counted as separate and equally valid; in other words, these are oriented meanders.
The values of T(n,4), n >= 4, form a series that increases by a multiplicative factor that gets closer and closer (alternating approaches from above and below) to a value of 4.4547 +/ 0.0007: 11, 42, 199, 858, 3881, 17156, 76707, 341060, 1520623, 6770556, 30165937, 134358958.
The values of T(n,5), n >= 5, form a series that increases by a multiplicative factor that gets closer and closer (alternating approaches from above and below) to a value of 9.421 +/ 0.014: 320, 3278, 29904, 285124, 2671052, 25200508, 237074534. (End)
It appears that T(n>=4,4) satisfies a recurrence with minimal polynomial x^6  7*x^5 + 7*x^4 + 10*x^3  9*x^2  3*x + 1; if so, then the ratio that T(n+1,4)/T(n,4) approaches as n goes to infinity is (1/12)*sqrt(24*sqrt(115)*cos((1/3)*Pi + (1/3)*arctan((3/1016)*sqrt(3)*sqrt(18097))) + 273) + (1/2)*sqrt((2/3)*sqrt(115)*cos((1/3)*Pi + (1/3)*arctan((3/1016)*sqrt(3)*sqrt(18097))) + (201/2)/sqrt(24*sqrt(115)*cos((1/3)*Pi + (1/3)*arctan((3/1016)*sqrt(3)*sqrt(18097))) + 273) + 91/6) + 3/4.  D. S. McNeil, Nov 30 2011


LINKS



FORMULA

T(n,3) is given by A078008, the expansion of (1x)/(1x2*x^2). Benoit Jubin noticed (Nov 22 2011) that T(n,3) is also given by 2*(b(n2) + b(n3) + b(n4) + ... +b(2)).


EXAMPLE

The 199 meanders on a 6 X 4 rectangle are shown in the supporting png image.


CROSSREFS

Cf. A200893, where the meanders on an n X k rectangle are unoriented, i.e., the sequence is reduced for symmetry.
Cf. A200749, which counts oriented meanders on an n X n square grid.
Cf. A200000, which counts unoriented meanders on an n X n square grid.


KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



