A200000 -id:A200000 - cp's OEIS frontend

A261400 Number of n X n knot mosaics.

1, 2, 22, 2594, 4183954, 101393411126, 38572794946976686, 234855052870954505606714, 23054099362200397056093750003442, 36564627559441095000442883434988307728126, 937273142571326346553334567317274833729462713413038

Offset: 1

N. J. A. Sloane, Aug 18 2015

nonn

According to Oh, Hong, Lee, and Lee, a(n) grows at a quadratic exponential rate. Moreover, it appears that the ratios A374947(n)/a(n) converge to 0 at a quadratic exponential rate. - Luc Ta, Aug 27 2024

Luc Ta, Table of n, a(n) for n = 1..14
K. Hong, H. Lee, H. J. Lee and S. Oh, Small knot mosaics and partition matrices, J. Phys. A: Math. Theor. 47 (2014) 435201; arXiv:1312.4009 [math.GT].
K. Hong, H. J. Lee, H. Lee and S. Oh, Upper bound on the total number of knot n-mosaics, J. Knot Theory Ramifications, Volume 23, Issue 13, November 2014; arXiv:1303.7044 [math.GT].
Hwa Jeong Lee, Kyungpyo Hong, Ho Lee, and Seungsang Oh, Mosaic number of knots, arXiv: 1301.6041 [math.GT], 2014.
Samuel J. Lomonaco and Louis H. Kauffman, Quantum Knots and Mosaics, Proc. Sympos. Applied Math., Amer. Math. Soc., Vol. 68 (2010), pp. 177-208.
Samuel J. Lomonaco and Louis H. Kauffman, Illustration for a(3) = 22, from "Quantum Knots and Mosaics", 2010, with permission.
Seungsang Oh, Kyungpyo Hong, Ho Lee, and Hwa Jeong Lee, Quantum knots and the number of knot mosaics, arXiv: 1412.4460 [math.GT], 2014.
Index entries for sequences related to knots

Reminiscent of (but of course different from) A200000.
The term 22 is the same 22 that appears in A261399.
a(n) is the main diagonal of A375353.
Cf. A374947, A375354, A375355, A375356, A375357.

x[0] = o[0] = {{1}};
x[n_] := ArrayFlatten[{{x[n - 1], o[n - 1]}, {o[n - 1], x[n - 1]}}];
o[n_] := ArrayFlatten[{{o[n - 1], x[n - 1]}, {x[n - 1], 4*o[n - 1]}}];
mosaicsSquare[n_] := If[n > 1, 2*Total[MatrixPower[x[n - 2] + o[n - 2], n - 2], 2], 1];
Flatten[ParallelTable[mosaicsSquare[n], {n, 1, 11}]] (* This program is based on Theorem 1 of Oh, Hong, Lee, and Lee (see Links). - Luc Ta, Aug 13 2024 *)

a(7)-a(11) from Hiroaki Yamanouchi, Aug 19 2015

Typo in a(11) corrected by Luc Ta, Aug 13 2024

A200893 Triangle read by rows: number of meanders filling out an n X k grid.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 1, 4, 0, 1, 1, 14, 42, 0, 1, 3, 63, 843, 9050, 0, 1, 3, 224, 7506, 342743, 6965359, 0, 1, 8, 1022, 71542, 6971973

Offset: 1

Jon Wild, Nov 23 2011

The sequence counts the distinct 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 not considered distinct.

			The 14 solutions for (n,k)=(5,4), 63 solutions for (n,k)=(6,4) and 224 solutions for (n,k)=(7,4) are illustrated in the supporting png files.

Cf. A200000 (sequence of entries for square grid).

T(n,3) appears to be equal to A090597.

A200749 Number of meanders filling out an n X n grid, not reduced for symmetry.

Original entry on oeis.org

1, 1, 0, 11, 320, 71648, 55717584, 213773992667, 3437213982024260, 249555807519163873078, 78627163663841340597702692, 109477494899001088619906813170744, 666376868834051436218404625691790011056, 17813932068751803215543399261217225231408150272, 2084618062581510894785237376608868017658716989948775752, 1069049587048126292657245511018395164729584995637677006604201633, 2399885835948485973061191866831331382214612321025714609065977840609754872

Offset: 1

Jon Wild, Nov 21 2011

nonn

The sequence counts the 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 counted separately.

			a(1) counts the paths that visit the single cell of the 1 X 1 lattice: there is one, the "fat dot".
The 11 solutions for n=4 are illustrated in the supporting .png file.

Jon Wild, Illustration for a(4) = 11.

A200000 gives the reduced version of the sequence (rotations/reflections not considered distinct).

a(8) - a(15) from Alex Chernov, Jan 01 2012

a(16) - a(17) from Zhao Hui Du, Apr 01 2014

A201145 Triangle read by rows: number of meanders filling out an n X k grid, unreduced for symmetry.

Original entry on oeis.org

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

Offset: 1

Jon Wild, Nov 27 2011

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.
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

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

Alex Chernov, Rows 1..15 of triangle, flattened
Alex Chernov, Some terms for rows above 15
Jon Wild, Illustration for T(6,4) = 199

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.

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)).

More terms from Alex Chernov, Jan 01 2012

cp's OEIS Frontend

A261400 Number of n X n knot mosaics.

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A200893 Triangle read by rows: number of meanders filling out an n X k grid.

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A200749 Number of meanders filling out an n X n grid, not reduced for symmetry.

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A201145 Triangle read by rows: number of meanders filling out an n X k grid, unreduced for symmetry.

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