OFFSET
2,1
COMMENTS
The formula has been proved.
REFERENCES
Yuanan Diao, Michael Lee Finney, Dawn Ray. The number of oriented rational links with a given deficiency number, Journal of Knot Theory and its Ramifications, Vol 30, Number 9, 2021. 2150065_1-20. See Theorem 4.3 and its proof.
LINKS
Paolo Xausa, Table of n, a(n) for n = 2..1000
Yuanan Diao, Michael Lee Finney, and Dawn Ray, The number of oriented rational links with a given deficiency number, arXiv:2007.02819 [math.GT], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,4,-2,-4).
FORMULA
a(n) = (2^(n-2) + (-1)^n*2)/3 + ((-1)^n+1)*2^((n-4)/2).
G.f.: 2*x^2*(1 - x - 2*x^2 + x^3)/((1 + x)^(1 - 2*x)*(1 - 2*x^2)). - Stefano Spezia, Mar 31 2022
EXAMPLE
If n=2 there are two rational links, namely, the Hopf link pair, one with positive crossings and the other with negative crossings. There are no two-component rational links with crossing number 3.
MATHEMATICA
LinearRecurrence[{1, 4, -2, -4}, {2, 0, 4, 2}, 50] (* Paolo Xausa, May 27 2024 *)
PROG
(PARI) a(n) = (2^(n-2) + (-1)^n*2)/3 + ((-1)^n+1)*2^((n-4)/2); \\ Michel Marcus, Mar 31 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Yuanan Diao, Mar 30 2022
STATUS
approved