A051449 - OEIS

%I #23 Jun 17 2022 14:44:24

%S 1,1,1,2,3,4,7,10,16,25,40,62,101,159,257,410,663,1062,1719,2764,4472,

%T 7209,11664,18828,30465,49221,79641,128746,208315,336872,545071,

%U 881638,1426520,2307665,3733880,6040746,9774133,15813587,25586921,41398418

%N Number of fibered rational knots with n crossings.

%H Harvey P. Dale, <a href="/A051449/b051449.txt">Table of n, a(n) for n = 3..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,2,1,-2,-2,-2,-1).

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

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

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

%t 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 *)

%t LinearRecurrence[{0,2,2,1,-2,-2,-2,-1},{1,1,1,2,3,4,7,10},40] (* _Harvey P. Dale_, Dec 27 2015 *)

%K easy,nonn,nice

%O 3,4

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

%E More terms from _James A. Sellers_