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 A375355 #17 Aug 26 2024 10:13:40
%S A375355 7,29,359,133,5519,316249,641,91283,19946891,4934695175,3157,1549799,
%T A375355 1298065813,1268810595131,1300161356831107,15689,26576579,85436799491,
%U A375355 330595705214327,1353434715973001999,5644698772550126097593,78253,457549079,5648174618317,86566215054880187,1416905739955631598043,23696846086162116561085541,399312236302057306354637147077
%N A375355 T(m, n) is the number of m X n period knot/link mosaics read by rows, with 1 <= n <= m.
%C A375355 An m X n mosaic is an m X n array of the 11 tiles given by Lomonaco and Kauffman. A period m X n mosaic is an m X n mosaic whose opposite edges are identified. A period mosaic depicts a knot or link iff the connection points of each tile coincide with those of the contiguous tiles and with those of the tiles on identified edges.
%C A375355 The Mathematica program below is based on the algorithm given in Theorem 2 of Oh, Hong, Lee, Lee, and Yeon.
%C A375355 T(m, n) >= A375356(m, n) for all m and n, with equality iff m = n = 1.
%C A375355 T(m, 1) = A074600(m) for all m. To see this, proceed by induction on m. In Theorem 2 of Oh, Hong, Lee, Lee, and Yeon, it is clear that tr(X_{m+1}) = 2*tr(X_m) and tr(O_{m+1}) = 5*tr(O_m) for all m. The theorem states that T(m+1, 1) = tr(X_{m+1} + O_{m+1}) = tr(X_{m+1}) + tr(O_{m+1}) = 2*tr(X_m) + 5*tr(O_m), and the claim follows since tr(X_1 + O_1) = 7.
%H A375355 Luc Ta, <a href="/A375355/b375355.txt">First 11 rows of the triangle, flattened</a>
%H A375355 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 A375355 Seungsang Oh, Kyungpyo Hong, Ho Lee, Hwa Jeong Lee, and Mi Jeong Yeon, <a href="http://arxiv.org/abs/1703.04867">Period and toroidal knot mosaics</a>, arXiv: 1703.04867 [math.GT], 2017.
%H A375355 <a href="/index/K#knots">Index entries for sequences related to knots</a>
%e A375355 Triangle begins:
%e A375355 7;
%e A375355 29, 359;
%e A375355 133, 5519, 316249;
%e A375355 641, 91283, 19946891, 4934695175;
%e A375355 3157, 1549799, 1298065813, 1268810595131, 1300161356831107;
%e A375355 ...
%e A375355 T(1,1) = 7 since the only period 1 X 1 link mosaics are given by the tiles T_0 and T_5 through T_10 of Lomonaco and Kauffman.
%t A375355 x[0] = o[0] = {{1}}; y[0] = p[0] = {{0}};
%t A375355 x[n_] := ArrayFlatten[{{x[n - 1], p[n - 1]}, {p[n - 1], x[n - 1]}}];
%t A375355 y[n_] := ArrayFlatten[{{y[n - 1], o[n - 1]}, {o[n - 1], y[n - 1]}}];
%t A375355 o[n_] := ArrayFlatten[{{o[n - 1], y[n - 1]}, {y[n - 1], 4 * o[n - 1]}}];
%t A375355 p[n_] := ArrayFlatten[{{p[n - 1], x[n - 1]}, {x[n - 1], 4 * p[n - 1]}}];
%t A375355 periodcount[m_, n_] := Tr[MatrixPower[x[m] + o[m], n]];
%t A375355 Flatten[ParallelTable[periodcount[m, n], {m, 1, 11}, {n, 1, m}]]
%Y A375355 Cf. A375356, A375357, A375353, A375354, A261400, A374947, A374946.
%K A375355 nonn,tabl
%O A375355 1,1
%A A375355 _Luc Ta_, Aug 20 2024