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 A321125 T(n,k) = b(n+k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0). 2
 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Let (A,B,d) denote the three-variable bracket polynomial for the two-bridge knot with Conway's notation C(n,k). Then T(n,k) is the leading coefficient of the reduced polynomial x*(1,1,x). In Kauffman's language, T(n,k) is the number of  states of the two-bridge knot C(n,k) corresponding to the maximum number of Jordan curves. REFERENCES Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983. LINKS Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened). Louis H. Kauffman, State models and the Jones polynomial, Topology Vol. 26 (1987), 395-407. Kelsey Lafferty, The three-variable bracket polynomial for reduced, alternating links, Rose-Hulman Undergraduate Mathematics Journal Vol. 14 (2013), 98-113. Matthew Overduin, The three-variable bracket polynomial for two-bridge knots, California State University REU, 2013. Franck Maminirina Ramaharo, Illustration of T(2,2) Franck Maminirina Ramaharo, Note on this sequence and related ones Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019. Eric Weisstein's World of Mathematics, Bracket Polynomial Wikipedia, 2-bridge knot Wikipedia, Bracket polynomial FORMULA T(n,0) = T(0,n) = 1, and T(n,k) = b(n+k) - b(n)*b(k) - b(n*k) + c(n)*c(k) for n >= 1, k >= 1, where b(n) = A154272(n+1) and c(n) = A294619(n). T(n,1) = A300453(n+1,A321126(n,1)). T(n,2) = A300454(n,A321126(n,2)). T(n,n) = A321127(n,A004280(n+1)). G.f.: (1 + (x - x^2)*y - (x - 3*x^2 + x^3)*y^2 - x^2*y^3)/((1 - x)*(1 - y)). E.g.f.: ((x^2 + 2*exp(x))*exp(y) - x^2 + (2*x - x^2)*y - (1 + x - exp(x))*y^2)/2. EXAMPLE Square array begins:   1, 1, 1, 1, 1, 1, ...   1, 2, 1, 1, 1, 1, ...   1, 1, 3, 2, 2, 2, ...   1, 1, 2, 1, 1, 1, ...   1, 1, 2, 1, 1, 1, ...   1, 1, 2, 1, 1, 1, ...   ... MATHEMATICA b[n_] = If[n == 0 || n == 2, 1, 0]; T[n_, k_] = b[n + k] - (2*b[n]*b[k] + 1)*b[n*k] + b[n] + b[k] + 1; Table[T[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten PROG (Maxima) b(n) := if n = 0 or n = 2 then 1 else 0\$ /* A154272(n+1) */ T(n, k) := b(n + k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1\$ create_list(T(k, n - k), n, 0, 12, k, 0, n); CROSSREFS Cf. A300453, A300454, A316989, A321126, A321127. Sequence in context: A326976 A117358 A294333 * A205617 A204112 A186027 Adjacent sequences:  A321122 A321123 A321124 * A321126 A321127 A321128 KEYWORD nonn,easy,tabl AUTHOR Franck Maminirina Ramaharo, Nov 24 2018 STATUS approved

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Last modified September 26 16:12 EDT 2021. Contains 347669 sequences. (Running on oeis4.)