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A374947 a(n) is the number of suitably connected Legendrian n-Mosaics.

%I A374947 #34 Oct 27 2024 12:12:33
%S A374947 1,2,20,1504,948032,5204262912,254112496082944,111879597850371293184,
%T A374947 448381477417976615986528256,16469260582635747355818375736459264,
%U A374947 5571666891811926168753521842383673521864704,17424018517043252553551626372130243982114254609186816
%N A374947 a(n) is the number of suitably connected Legendrian n-Mosaics.
%C A374947 A Legendrian n-mosaic is an n X n array of the 10 tiles given in Figure 5 of Pezzimenti and Pandey. These tiles represent part of a Legendrian curve in the front projection.
%C A374947 A Legendrian n-mosaic is suitably connected iff the connection points of each tile coincide with those of all contiguous tiles. Note that the n-mosaic consisting of all blank tiles is vacuously suitably connected even though it does not represent a link.
%C A374947 This is the main diagonal of A375354. It appears to grow at a quadratic exponential rate, and the ratios a(n)/A261400(n) seem to converge to 0 at a quadratic exponential rate.
%C A374947 For more information, see Sections 4 and 5 of Kipe et al. In particular, see Figures 20 and 21 for explicit best-fit models. - _Luc Ta_, Oct 27 2024
%H A374947 Margaret Kipe, <a href="/A374947/a374947.rs.txt">Rust</a>
%H A374947 Margaret Kipe, Samantha Pezzimenti, Leif Schaumann, Luc Ta, and Wing Hong Tony Wong, <a href="http://arxiv.org/abs/2410.08064">Bounds on the mosaic number of Legendrian knots</a>, arXiv: 2410.08064 [math.GT], 2024.
%H A374947 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 A374947 S. Pezzimenti and A. Pandey, <a href="https://doi.org/10.1142/S021821652250002X">Geography of Legendrian knot mosaics</a>, Journal of Knot Theory and its Ramifications, 31 (2022), article no. 2250002, 1-22.
%H A374947 <a href="/index/K#knots">Index entries for sequences related to knots</a>
%e A374947 For n = 2 there are exactly a(2) = 2 suitably connected Legendrian 2-mosaics, namely the empty mosaic and the Legendrian unknot with maximal Thurston-Bennequin invariant.
%t A374947 x[0] = o[0] = {{1}};
%t A374947 x[n_] := ArrayFlatten[{{x[n - 1], o[n - 1]}, {o[n - 1], x[n - 1]}}];
%t A374947 o[n_] := ArrayFlatten[{{o[n - 1], x[n - 1]}, {x[n - 1], 3*o[n - 1]}}];
%t A374947 legendrianSquare[n_] := If[n > 1, 2*Total[MatrixPower[x[n - 2] + o[n - 2], n - 2], 2], 1];
%t A374947 Flatten[ParallelTable[legendrianSquare[n], {n, 1, 11}]] (* This program is adapted from Theorem 1 of Oh, Hong, Lee, and Lee (see Links, cf. A375354). - _Luc Ta_, Aug 20 2024 *)
%o A374947 (Rust) // See Margaret Kipe link
%Y A374947 Cf. A261400, A375354, A374939, A374942, A374943, A374944, A374945, A374946, A375353, A375355, A375356, A375357.
%K A374947 nonn
%O A374947 1,2
%A A374947 _Margaret Kipe_, _Samantha Pezzimenti_, _Leif Schaumann_, _Luc Ta_, and _Wing Hong Tony Wong_, Jul 24 2024
%E A374947 a(7)-a(11) from _Luc Ta_, Aug 20 2024
%E A374947 a(12) from _Alois P. Heinz_, Aug 20 2024