|
%I #48 Mar 20 2024 19:15:53
%S 1,1,2,3,7,12,24,45,91,176,352,693,1387,2752,5504,10965,21931,43776,
%T 87552,174933,349867,699392,1398784,2796885,5593771,11186176,22372352,
%U 44741973,89483947,178962432,357924864,715838805,1431677611,2863333376,5726666752
%N Number of rational knots (or two-bridge knots) with n crossings (up to mirroring).
%D S. Jablan and R. Sazdanović, LinKnot: Knot Theory by Computer, World Scientific Press, 2007.
%H Vincenzo Librandi, <a href="/A018240/b018240.txt">Table of n, a(n) for n = 3..1000</a>
%H C. Ernst and D. W. Sumners, <a href="https://doi.org/10.1017/S0305004100067323">The Growth of the Number of Prime Knots</a>, Math. Proc. Cambridge Philos. Soc. 102, 303-315, 1987 (see Theorem 5, formulas for TK_n).
%H Taizo Kanenobu and Toshio Sumi, <a href="https://doi.org/10.1090/S0025-5718-1993-1176711-4">Polynomial Invariants of 2-Bridge Knots through 22 Crossings</a>, Math. Comp. 60 (1993), 771-778, S17 (see Table 2).
%H P.-V. Koseleff, D. Pecker, <a href="https://arxiv.org/abs/1011.5992">Conway polynomials of two-bridge links</a>, arXiv:1011.5992 [math.GT], 2010-2012 (only version 1 contains tables).
%H P.-V. Koseleff, D. Pecker, <a href="https://doi.org/10.1016/j.jsc.2014.09.011">On Alexander-Conway polynomials of two-bridge links</a>, Journal of Symbolic Computation 68 (2015), 215-229.
%H A. Stoimenow, <a href="https://doi.org/10.1016/j.jalgebra.2006.11.031">Generating functions, Fibonacci numbers and rational knots</a>, Journal of Algebra, 310 (2007), 491-525.
%H <a href="/index/K#knots">Index entries for sequences related to knots</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (-1,5,5,-2,-2,-8,-8).
%F a(n) = - a(n-1) + 5*(a(n-2)+a(n-3)) - 2*(a(n-4)+a(n-5)) - 8*(a(n-6)+a(n-7)). [Originally contributed as a separate sequence entry by _Thomas A. Gittings_, Dec 11 2003; see Stoimenow, Corollary 5.1 for proof]
%F G.f.: (1-2*x^2-x^3-x^4)*x^3/((1-2*x)*(1+x)*(1-2*x^2)*(1+x^2)). - _R. J. Mathar_, Sep 08 2008
%e The a(7)=7 rational knots with 7 crossings are 7, 52, 43, 322, 313, 2212, 21112. All the rational knots are listed in A122495.
%t LinearRecurrence[{-1, 5, 5, -2, -2, -8, -8}, {1, 1, 2, 3, 7, 12, 24}, 50] (* _Harvey P. Dale_, Sep 03 2013 *)
%t CoefficientList[Series[(1 - 2 x^2 - x^3 - x^4)/((1 - 2 x) (1 + x) (1 - 2 x^2) (1 + x^2)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 07 2014 *)
%o (PARI) Vec((1-2*x^2-x^3-x^4)*x^3/((1-2*x)*(1+x)*(1-2*x^2)*(1+x^2))+O(x^66)) \\ _Joerg Arndt_, Aug 07 2014
%Y Cf. A122495, A005418, A002863, A086825, A089790.
%Y Cf. A018240 = number of rational knots, A005418 = number of rational knots and links, A001045 = Jacobsthal sequence (the difference between the number of rational links and knots), A090597 = rational links with n crossings, A329908, A336398.
%K nice,easy,nonn
%O 3,3
%A Alexander Stoimenow (stoimeno(AT)math.toronto.edu)
%E Edited by _Andrey Zabolotskiy_, Jun 18 2020
|
