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

2,1

Links

Table of n, a(n) for n=2..36.

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).

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

Prog

(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));

Crossrefs

Cf. A018240, A090597, A336398.

Sequence in context: A345470 A368727 A320296 * A295198 A080880 A305606

Adjacent sequences: A329905 A329906 A329907 * A329909 A329910 A329911

Keyword

nonn,easy

Author

Michel Marcus, Jul 07 2020

Status

approved