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A201145
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Triangle read by rows: number of meanders filling out an n by k grid, unreduced for symmetry.
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3
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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
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1,9
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COMMENTS
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The sequence counts the closed paths that visit every cell of an n-by-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.
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.
(preceding comments by J. Wild, Nov 29 2011)
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]
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LINKS
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Alex Chernov, Rows 1..15 of triangle, flattened
Jon Wild, Illustration for T(6,4) = 199
Alex Chernov, Some terms for rows above 15
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FORMULA
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T(n,3) is given by A078008, the expansion of (1-x)/(1-x-2*x^2). Benoît Jubin noticed (22 Nov 2011) that T(n,3) is also given by 2*(b(n-2)+b(n-3)+b(n-4)....+b(2)).
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EXAMPLE
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The 199 meanders on a 6X4 rectangle are shown in the supporting png image.
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CROSSREFS
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Cf. A200893, where the meanders on an nXk rectangle are unoriented, i.e. the sequence is reduced for symmetry.
Cf. A200749, which counts oriented meanders on an nXn square grid.
Cf. A200000, which counts unoriented meanders on an nXn square grid.
Sequence in context: A057095 A119189 A202952 * A053994 A057213 A224480
Adjacent sequences: A201142 A201143 A201144 * A201146 A201147 A201148
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KEYWORD
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nonn,tabl
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AUTHOR
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Jon Wild, Nov 27 2011
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EXTENSIONS
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More terms from Alex Chernov, Jan 01 2012
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STATUS
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approved
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