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

1,9

COMMENTS

The sequence counts the closed paths that visit every cell of an n X k rectangular 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 counted as separate and equally valid; in other words these are oriented meanders.

From Jon Wild, Nov 29 2011: (Start)

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

Alex Chernov, Rows 1..15 of triangle, flattened

Jon Wild, Illustration for T(6,4) = 199

Alex Chernov, Some terms for rows above 15

FORMULA

T(n,3) is given by A078008, the expansion of (1-x)/(1-x-2*x^2). Benoit Jubin noticed (Nov 22 2011) that T(n,3) is also given by 2*(b(n-2)+b(n-3)+b(n-4)....+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.

Sequence in context: A057095 A119189 A202952 * A053994 A057213 A224480

Adjacent sequences:  A201142 A201143 A201144 * A201146 A201147 A201148

KEYWORD

nonn,tabl

AUTHOR

Jon Wild, Nov 27 2011

EXTENSIONS

More terms from Alex Chernov, Jan 01 2012

STATUS

approved

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Last modified April 20 13:42 EDT 2014. Contains 240806 sequences.