cp's OEIS Frontend

A261400 Number of n X n knot mosaics.

%I A261400 #45 Aug 28 2024 03:36:59
%S A261400 1,2,22,2594,4183954,101393411126,38572794946976686,
%T A261400 234855052870954505606714,23054099362200397056093750003442,
%U A261400 36564627559441095000442883434988307728126,937273142571326346553334567317274833729462713413038
%N A261400 Number of n X n knot mosaics.
%C A261400 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
%H A261400 Luc Ta, <a href="/A261400/b261400.txt">Table of n, a(n) for n = 1..14</a>
%H A261400 K. Hong, H. Lee, H. J. Lee and S. Oh, <a href="http://arxiv.org/abs/1312.4009">Small knot mosaics and partition matrices</a>, J. Phys.  A: Math. Theor. 47 (2014) 435201; arXiv:1312.4009 [math.GT].
%H A261400 K. Hong, H. J. Lee, H. Lee and S. Oh, <a href="http://arxiv.org/abs/1303.7044">Upper bound on the total number of knot n-mosaics</a>, J. Knot Theory Ramifications, Volume 23, Issue 13, November 2014; arXiv:1303.7044 [math.GT].
%H A261400 Hwa Jeong Lee, Kyungpyo Hong, Ho Lee, and Seungsang Oh, <a href="http://arxiv.org/abs/1301.6041">Mosaic number of knots</a>, arXiv: 1301.6041 [math.GT], 2014.
%H A261400 Samuel J. Lomonaco and Louis H. Kauffman, <a href="http://www.csee.umbc.edu/~lomonaco/pubs/psapm561.pdf">Quantum Knots and Mosaics</a>, Proc. Sympos. Applied Math., Amer. Math. Soc., Vol. 68 (2010), pp. 177-208.
%H A261400 Samuel J. Lomonaco and Louis H. Kauffman, <a href="/A261400/a261400.pdf">Illustration for a(3) = 22</a>, from "Quantum Knots and Mosaics", 2010, with permission.
%H A261400 Seungsang Oh, Kyungpyo Hong, Ho Lee, and Hwa Jeong Lee, <a href="http://arxiv.org/abs/1412.4460">Quantum knots and the number of knot mosaics</a>, arXiv: 1412.4460 [math.GT], 2014.
%H A261400 <a href="/index/K#knots">Index entries for sequences related to knots</a>
%t A261400 x[0] = o[0] = {{1}};
%t A261400 x[n_] := ArrayFlatten[{{x[n - 1], o[n - 1]}, {o[n - 1], x[n - 1]}}];
%t A261400 o[n_] := ArrayFlatten[{{o[n - 1], x[n - 1]}, {x[n - 1], 4*o[n - 1]}}];
%t A261400 mosaicsSquare[n_] := If[n > 1, 2*Total[MatrixPower[x[n - 2] + o[n - 2], n - 2], 2], 1];
%t A261400 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 *)
%Y A261400 Reminiscent of (but of course different from) A200000.
%Y A261400 The term 22 is the same 22 that appears in A261399.
%Y A261400 a(n) is the main diagonal of A375353.
%Y A261400 Cf. A374947, A375354, A375355, A375356, A375357.
%K A261400 nonn
%O A261400 1,2
%A A261400 _N. J. A. Sloane_, Aug 18 2015
%E A261400 a(7)-a(11) from _Hiroaki Yamanouchi_, Aug 19 2015
%E A261400 Typo in a(11) corrected by _Luc Ta_, Aug 13 2024