A018240 Number of rational knots (or two-bridge knots) with n crossings (up to mirroring).

1, 1, 2, 3, 7, 12, 24, 45, 91, 176, 352, 693, 1387, 2752, 5504, 10965, 21931, 43776, 87552, 174933, 349867, 699392, 1398784, 2796885, 5593771, 11186176, 22372352, 44741973, 89483947, 178962432, 357924864, 715838805, 1431677611, 2863333376, 5726666752

Offset: 3

Author

Alexander Stoimenow (stoimeno(AT)math.toronto.edu)

Keywords

• nice
• easy
• nonn

Examples

			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.
		

References

• S. Jablan and R. Sazdanović, LinKnot: Knot Theory by Computer, World Scientific Press, 2007.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 3..1000
• C. Ernst and D. W. Sumners, The Growth of the Number of Prime Knots, Math. Proc. Cambridge Philos. Soc. 102, 303-315, 1987 (see Theorem 5, formulas for TK_n).
• Taizo Kanenobu and Toshio Sumi, Polynomial Invariants of 2-Bridge Knots through 22 Crossings, Math. Comp. 60 (1993), 771-778, S17 (see Table 2).
• P.-V. Koseleff, D. Pecker, Conway polynomials of two-bridge links, arXiv:1011.5992 [math.GT], 2010-2012 (only version 1 contains tables).
• P.-V. Koseleff, D. Pecker, On Alexander-Conway polynomials of two-bridge links, Journal of Symbolic Computation 68 (2015), 215-229.
• A. Stoimenow, Generating functions, Fibonacci numbers and rational knots, Journal of Algebra, 310 (2007), 491-525.
• Index entries for sequences related to knots
• Index entries for linear recurrences with constant coefficients, signature (-1,5,5,-2,-2,-8,-8).

Crossrefs

Cf. A122495, A005418, A002863, A086825, A089790.

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.

Programs

Mathematica

LinearRecurrence[{-1, 5, 5, -2, -2, -8, -8}, {1, 1, 2, 3, 7, 12, 24}, 50] (* Harvey P. Dale, Sep 03 2013 *)
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 *)

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

Formula

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]

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

Extensions

Edited by Andrey Zabolotskiy, Jun 18 2020