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 A321126 #17 Feb 16 2025 08:33:57
%S A321126 1,2,2,3,2,3,4,3,3,4,5,4,3,4,5,6,5,4,4,5,6,7,6,5,5,5,6,7,8,7,6,6,6,6,
%T A321126 7,8,9,8,7,7,7,7,7,8,9,10,9,8,8,8,8,8,8,9,10,11,10,9,9,9,9,9,9,9,10,
%U A321126 11,12,11,10,10,10,10,10,10,10,10,11,12,13,12,11,11,11,11,11,11,11,11,11,12,13
%N A321126 T(n,k) = max(n + k - 1, n + 1, k + 1), square array read by antidiagonals (n >= 0, k >= 0).
%C A321126 T(n,k) - 1 is the maximum degree of d in the three-variable bracket polynomial <K>(A,B,d) for the two-bridge knot with Conway's notation C(n,k). Hence, T(n,k) is the maximum number of Jordan curves that are obtained by splitting the crossings of such knot diagram.
%D A321126 Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.
%H A321126 Michael De Vlieger, <a href="/A321126/b321126.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened).
%H A321126 Louis H. Kauffman, <a href="https://doi.org/10.1016/0040-9383(87)90009-7">State models and the Jones polynomial</a>, Topology Vol. 26 (1987), 395-407.
%H A321126 Kelsey Lafferty, <a href="https://scholar.rose-hulman.edu/rhumj/vol14/iss2/7/">The three-variable bracket polynomial for reduced, alternating links</a>, Rose-Hulman Undergraduate Mathematics Journal Vol. 14 (2013), 98-113.
%H A321126 Matthew Overduin, <a href="https://www.math.csusb.edu/reu/OverduinPaper.pdf">The three-variable bracket polynomial for two-bridge knots</a>, California State University REU, 2013.
%H A321126 Franck Maminirina Ramaharo, <a href="/A321125/a321125.pdf">Illustration of T(2,2)</a>
%H A321126 Franck Maminirina Ramaharo, <a href="/A321125/a321125.txt">Note on sequence A321125 and related ones</a>
%H A321126 Franck Ramaharo, <a href="https://arxiv.org/abs/1902.08989">A generating polynomial for the two-bridge knot with Conway's notation C(n,r)</a>, arXiv:1902.08989 [math.CO], 2019.
%H A321126 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BracketPolynomial.html">Bracket Polynomial</a>
%H A321126 Wikipedia, <a href="https://en.wikipedia.org/wiki/2-bridge_knot">2-bridge knot</a>
%H A321126 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bracket_polynomial">Bracket polynomial</a>
%F A321126 T(n,k) = T(k,n).
%F A321126 T(n,k) = A051125(n+1,k+1) for 0 <= k <= 2, n >= 0, and T(n,k) = A051125(n+1,k+1) + A003983(n-2,k-2) for k >= 3, n >= 3.
%F A321126 T(n,n) = A004280(n+1).
%F A321126 G.f.: (1 - (2*x - x^2)*y + (x - 2*x^2 + x^3)*y^2 + (x^2 - x^3)*y^3)/(((1 - x)*(1 - y))^2).
%e A321126 Square array begins:
%e A321126 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
%e A321126 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
%e A321126 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, ...
%e A321126 4, 4, 4, 5, 6, 7, 8, 9, 10, 11, ...
%e A321126 5, 5, 5, 6, 7, 8, 9, 10, 11, 12, ...
%e A321126 6, 6, 6, 7, 8, 9, 10, 11, 12, 13, ...
%e A321126 7, 7, 7, 8, 9, 10, 11, 12, 13, 14, ...
%e A321126 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, ...
%e A321126 9, 9, 9, 10, 11, 12, 13, 14, 15, 16, ...
%e A321126 10, 10, 10, 11, 12, 13, 14, 15, 16, 17, ...
%e A321126 ...
%t A321126 Table[Max[k + 1, n - 1, n - k + 1], {n, 0, 10}, {k, 0, n}] // Flatten
%o A321126 (Maxima) create_list(max(k + 1, n - 1, n - k + 1), n, 0, 10, k, 0, n);
%Y A321126 T(n,1) = degree of the (n+1)-th row polynomial in A300453.
%Y A321126 T(n,k) = degree of the n-th row polynomials in A300454 and A321127, k = 2,n, respectively.
%Y A321126 Cf. A321125, A316989.
%Y A321126 Cf. A003056, A003983, A051125.
%K A321126 nonn,tabl,easy
%O A321126 0,2
%A A321126 _Franck Maminirina Ramaharo_, Nov 21 2018