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A018240 Number of rational knots (or two-bridge knots) with n crossings (up to mirroring). 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,3

REFERENCES

Jablan S. and Sazdanovic R., 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 linear recurrences with constant coefficients, signature (-1,5,5,-2,-2,-8,-8).

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]

a(n) = {2^{n-3}+2^{[n/2]-2^{n (mod 2)}+ {[n/2] (mod 2)}(-1)^{n-1}}/3. - Slavik Jablan, Dec 20 2003

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

EXAMPLE

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.

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 *)

PROG

(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

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.

Sequence in context: A036538 A341407 A108742 * A090596 A355385 A321838

Adjacent sequences:  A018237 A018238 A018239 * A018241 A018242 A018243

KEYWORD

nice,easy,nonn

AUTHOR

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

EXTENSIONS

Edited by Andrey Zabolotskiy, Jun 18 2020

STATUS

approved

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Last modified September 25 12:12 EDT 2022. Contains 356984 sequences. (Running on oeis4.)