This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A375353 #21 Sep 07 2024 16:03:40
%S A375353 1,1,2,1,4,22,1,8,130,2594,1,16,778,54226,4183954,1,32,4666,1144526,
%T A375353 331745962,101393411126,1,64,27994,24204022,26492828950,
%U A375353 31507552821550,38572794946976686,1,128,167962,512057546,2119630825150,9841277889785426,47696523856560453790,234855052870954505606714
%N A375353 T(m,n) = Number of m X n knot/link mosaics read by rows, with 1<=n<=m.
%C A375353 An m X n link mosaic is a suitably connected m X n array of the 11 tiles given by Lomonaco and Kauffman. The condition of being suitably connected means that the connection points of each tile coincide with those of the contiguous tiles. Thus, link mosaics depict projections of a link or a knot onto a plane.
%C A375353 The Mathematica program below is based on the algorithm given in Theorem 1 of Oh, Hong, Lee, and Lee.
%H A375353 Luc Ta, <a href="/A375353/b375353.txt">First 11 rows of the triangle, flattened</a>
%H A375353 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], 2013-2014.
%H A375353 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 A375353 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 A375353 <a href="/index/K#knots">Index entries for sequences related to knots</a>
%F A375353 T(m,2) = A000079(m-1) for all m >= 2 and T(m,3) = A261399(m) for all m >= 3 due to Corollary 2 of Hong, H. Lee, H. J. Lee, and Oh.
%e A375353 Triangle begins:
%e A375353 1;
%e A375353 1, 2;
%e A375353 1, 4, 22;
%e A375353 1, 8, 130, 2594;
%e A375353 1, 16, 778, 54226, 4183954;
%e A375353 1, 32, 4666, 1144526, 331745962, 101393411126;
%e A375353 ...
%e A375353 T(2,2) = 2 since the only suitably connected 2 X 2 link mosaics are the empty mosaic and the mosaic depicting an unknot attaining its minimal crossing number.
%e A375353 For all n >= 1, we have T(n,1) = 1 since the only suitably connected mosaic with one column is empty.
%t A375353 x[0] = o[0] = {{1}};
%t A375353 x[n_] := ArrayFlatten[{{x[n - 1], o[n - 1]}, {o[n - 1], x[n - 1]}}];
%t A375353 o[n_] := ArrayFlatten[{{o[n - 1], x[n - 1]}, {x[n - 1], 4*o[n - 1]}}];
%t A375353 mosaics[m_, n_] := If[m > 1 && n > 1, 2*Total[MatrixPower[x[m - 2] + o[m - 2], n - 2], 2], 1];
%t A375353 Flatten[ParallelTable[mosaics[m, n], {m, 1, 11}, {n, 1, m}]] (* _Luc Ta_, Aug 13 2024 *)
%Y A375353 The main diagonal T(n,n) is A261400.
%Y A375353 Cf. A375354, A374947, A374946, A375355, A375356, A375357.
%K A375353 nonn,tabl
%O A375353 1,3
%A A375353 _Luc Ta_, Aug 13 2024