A329908 Number of oriented rational links with crossing number n.
2, 2, 5, 6, 15, 24, 51, 90, 187, 352, 715, 1386, 2795, 5504, 11051, 21930, 43947, 87552, 175275, 349866, 700075, 1398784, 2798251, 5593770, 11188907, 22372352, 44747435, 89483946, 178973355, 357924864, 715860651, 1431677610, 2863377067, 5726666752, 11453377195
Offset: 2
Links
- Yuanan Diao, Michael Finney, and Dawn Ray, The number of oriented rational links with a given deficiency number, arXiv:2007.02819 [math.GT], 2020. See Theorem 3 p.9 and Table 1 p. 14.
- C. Ernst and D. W. Sumners, The Growth of the Number of Prime Knots, Math. Proc. Cambridge Philos. Soc. 102, 303-315, 1987 (see Theorem 1, formulas for TL_n^*).
- Index entries for linear recurrences with constant coefficients, signature (1,3,-1,0,-2,-4).
Programs
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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));
Formula
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.
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
From Wesley Ivan Hurt, Jul 17 2025: (Start)
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.
a(n) = a(n-1) + 3*a(n-2) - a(n-3) - 2*a(n-5) - 4*a(n-6). (End)