A261400 Number of n X n knot mosaics.

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

Offset: 1

Author

N. J. A. Sloane, Aug 18 2015

Keywords

• nonn

Comments

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

Links

• 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

Crossrefs

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.

Programs

Mathematica

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 *)

Extensions

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

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