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%I #36 Jan 21 2023 02:15:15
%S 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,
%T 29904,1369736,55717584,0,1,22,3881,285124,27876028,2372510658,
%U 213773992667,0,1,42,17156,2671052,549405072,98927211122,18677872557034,3437213982024260
%N Triangle read by rows: number of meanders filling out an n X k grid, unreduced for symmetry.
%C 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 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.
%C From _Jon Wild_, Nov 29 2011: (Start)
%C 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.
%C 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)
%C 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
%H Alex Chernov, <a href="/A201145/b201145.txt">Rows 1..15 of triangle, flattened</a>
%H Alex Chernov, <a href="http://alex-black.ru/article.php?content=161">Some terms for rows above 15</a>
%H Jon Wild, <a href="/A201145/a201145.png">Illustration for T(6,4) = 199</a>
%F 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)).
%e The 199 meanders on a 6 X 4 rectangle are shown in the supporting png image.
%Y Cf. A200893, where the meanders on an n X k rectangle are unoriented, i.e., the sequence is reduced for symmetry.
%Y Cf. A200749, which counts oriented meanders on an n X n square grid.
%Y Cf. A200000, which counts unoriented meanders on an n X n square grid.
%K nonn,tabl
%O 1,9
%A _Jon Wild_, Nov 27 2011
%E More terms from _Alex Chernov_, Jan 01 2012
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