A352737 Number of oriented two-component rational links (or two-bridge links) with crossing number n (a chiral pair is counted as two distinct ones).
2, 0, 4, 2, 10, 10, 30, 42, 102, 170, 374, 682, 1430, 2730, 5590, 10922, 22102, 43690, 87894, 174762, 350550, 699050, 1400150, 2796202, 5596502, 11184810, 22377814, 44739242, 89494870, 178956970, 357946710, 715827882, 1431721302, 2863311530, 5726754134, 11453246122
Offset: 2
Examples
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.
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).
Programs
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Mathematica
LinearRecurrence[{1, 4, -2, -4}, {2, 0, 4, 2}, 50] (* Paolo Xausa, May 27 2024 *) -
PARI
a(n) = (2^(n-2) + (-1)^n*2)/3 + ((-1)^n+1)*2^((n-4)/2); \\ Michel Marcus, Mar 31 2022
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
Comments