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A329908 Number of oriented rational links with crossing number n.

%I A329908 #62 Jul 17 2025 00:07:15
%S A329908 2,2,5,6,15,24,51,90,187,352,715,1386,2795,5504,11051,21930,43947,
%T A329908 87552,175275,349866,700075,1398784,2798251,5593770,11188907,22372352,
%U A329908 44747435,89483946,178973355,357924864,715860651,1431677610,2863377067,5726666752,11453377195
%N A329908 Number of oriented rational links with crossing number n.
%H A329908 Yuanan Diao, Michael 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. See Theorem 3 p.9 and Table 1 p. 14.
%H A329908 C. Ernst and D. W. Sumners, <a href="https://doi.org/10.1017/S0305004100067323">The Growth of the Number of Prime Knots</a>, Math. Proc. Cambridge Philos. Soc. 102, 303-315, 1987 (see Theorem 1, formulas for TL_n^*).
%H A329908 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-1,0,-2,-4).
%F A329908 a(n) = (2^(n-1)+1)/3 + 2^(n/2-1) if n is even; (2^(n-1)+2^((n-1)/2)-2)/3 if n is odd and n == 1 mod 4; (2^(n-1)+2^((n-1)/2))/3 if n is odd and n == 3 mod 4.
%F A329908 G.f.: x^2*(2 - 3*x^2 - 3*x^3 - 4*x^4)/(1 - x - 3*x^2 + x^3 + 2*x^5 + 4*x^6). - _Jinyuan Wang_, Jul 08 2020
%F A329908 From _Wesley Ivan Hurt_, Jul 17 2025: (Start)
%F A329908 a(n) = (2^(n+1)+2^(n/2)*(3+sqrt(2)+(-1)^n*(3-sqrt(2)))+4*((-1)^n+sin(3*n*Pi/2)))/12.
%F A329908 a(n) = a(n-1) + 3*a(n-2) - a(n-3) - 2*a(n-5) - 4*a(n-6). (End)
%o A329908 (PARI) a(n) = if (n%2, if ((n%4)==1, (2^(n-1)+2^((n-1)/2)-2)/3, (2^(n-1)+2^((n-1)/2))/3), (2^(n-1)+1)/3 + 2^(n/2-1));
%Y A329908 Cf. A018240, A090597, A336398.
%K A329908 nonn,easy
%O A329908 2,1
%A A329908 _Michel Marcus_, Jul 07 2020