A051449 Number of fibered rational knots with n crossings.

1, 1, 1, 2, 3, 4, 7, 10, 16, 25, 40, 62, 101, 159, 257, 410, 663, 1062, 1719, 2764, 4472, 7209, 11664, 18828, 30465, 49221, 79641, 128746, 208315, 336872, 545071, 881638, 1426520, 2307665, 3733880, 6040746, 9774133, 15813587, 25586921, 41398418

Offset: 3

Author

Alexander Stoimenow (stoimeno(AT)math.toronto.edu)

Keywords

• easy
• nonn
• nice

Examples

			a(7)=3 because there are 3 fibered rational knots with 7 crossings: 7_1, 7_6 and 7_7 (in Alexander-Briggs notation).
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 3..1000
• Index entries for linear recurrences with constant coefficients, signature (0,2,2,1,-2,-2,-2,-1).

Programs

Mathematica

f[x_] = -(x+1)^2*(x^3+x-1) / ((x^2+x-1)*(x^2+x+1)*(x^4+x^2-1)); CoefficientList[ Series[f[x], {x, 0, 39}], x]; Table[a[n], {n, 0, 20}](* Jean-François Alcover, Nov 21 2011 *)
LinearRecurrence[{0,2,2,1,-2,-2,-2,-1},{1,1,1,2,3,4,7,10},40] (* Harvey P. Dale, Dec 27 2015 *)

Formula

G.f.: (x^2/2)*((-x-x^2)/(x^4+2x^3+x^2-1) + (-x-x^2)/(x^4+x^2-1)).

G.f.: -x^3*(x^3+x-1)*(1+x)^2 / ( (1+x+x^2)*(x^2+x-1)*(x^4+x^2-1) ).

Extensions

More terms from James Sellers