|
|
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
|
S. Jablan and R. Sazdanović, LinKnot: Knot Theory by Computer, World Scientific Press, 2007.
|
|
LINKS
|
|
|
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
|
|
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. 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.
|
|
KEYWORD
|
nice,easy,nonn
|
|
AUTHOR
|
Alexander Stoimenow (stoimeno(AT)math.toronto.edu)
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|