A352737 - OEIS

%I #21 May 27 2024 15:46:04

%S 2,0,4,2,10,10,30,42,102,170,374,682,1430,2730,5590,10922,22102,43690,

%T 87894,174762,350550,699050,1400150,2796202,5596502,11184810,22377814,

%U 44739242,89494870,178956970,357946710,715827882,1431721302,2863311530,5726754134,11453246122

%N Number of oriented two-component rational links (or two-bridge links) with crossing number n (a chiral pair is counted as two distinct ones).

%C The formula has been proved.

%D 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.

%H Paolo Xausa, <a href="/A352737/b352737.txt">Table of n, a(n) for n = 2..1000</a>

%H Yuanan Diao, Michael Lee Finney, and Dawn Ray, <a href="https://arxiv.org/abs/2007.02819">The number of oriented rational links with a given deficiency number</a>, arXiv:2007.02819 [math.GT], 2020.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-2,-4).

%F a(n) = (2^(n-2) + (-1)^n*2)/3 + ((-1)^n+1)*2^((n-4)/2).

%F 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

%e 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.

%t LinearRecurrence[{1, 4, -2, -4}, {2, 0, 4, 2}, 50] (* _Paolo Xausa_, May 27 2024 *)

%o (PARI) a(n) = (2^(n-2) + (-1)^n*2)/3 + ((-1)^n+1)*2^((n-4)/2); \\ _Michel Marcus_, Mar 31 2022

%Y Cf. A336398, A336030.

%K nonn,easy

%O 2,1

%A _Yuanan Diao_, Mar 30 2022