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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).

%I A352737 #21 May 27 2024 15:46:04
%S A352737 2,0,4,2,10,10,30,42,102,170,374,682,1430,2730,5590,10922,22102,43690,
%T A352737 87894,174762,350550,699050,1400150,2796202,5596502,11184810,22377814,
%U A352737 44739242,89494870,178956970,357946710,715827882,1431721302,2863311530,5726754134,11453246122
%N 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).
%C A352737 The formula has been proved.
%D A352737 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 A352737 Paolo Xausa, <a href="/A352737/b352737.txt">Table of n, a(n) for n = 2..1000</a>
%H A352737 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 A352737 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-2,-4).
%F A352737 a(n) = (2^(n-2) + (-1)^n*2)/3 + ((-1)^n+1)*2^((n-4)/2).
%F A352737 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 A352737 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 A352737 LinearRecurrence[{1, 4, -2, -4}, {2, 0, 4, 2}, 50] (* _Paolo Xausa_, May 27 2024 *)
%o A352737 (PARI) a(n) = (2^(n-2) + (-1)^n*2)/3 + ((-1)^n+1)*2^((n-4)/2); \\ _Michel Marcus_, Mar 31 2022
%Y A352737 Cf. A336398, A336030.
%K A352737 nonn,easy
%O A352737 2,1
%A A352737 _Yuanan Diao_, Mar 30 2022