A078477 Number of rational knots with n crossings and unknotting number = 1 (chiral pairs counted only once).

1, 1, 1, 3, 3, 6, 7, 15, 15, 30, 31, 63, 63, 126, 127, 255, 255, 510, 511, 1023, 1023, 2046, 2047, 4095, 4095, 8190, 8191, 16383, 16383, 32766, 32767, 65535, 65535, 131070, 131071, 262143, 262143, 524286, 524287, 1048575, 1048575, 2097150, 2097151

Offset: 3

Author

Ralf Stephan, Jan 03 2003

Keywords

• nonn
• easy

Comments

From Alexander Adamchuk, Nov 16 2009: (Start)

For n>1 a(2n+1) = 2^(n-1) - 1 = A000225(n-1).

For n>1 a(4n) = a(4n+1) - 1 = 2^(2n-1) - 2.

For n>0 a(4n+2) = a(4n+3) = 2^(2n) - 1. (End)

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 3..1000
• A. Stoimenow, Generating Functions, Fibonacci Numbers and Rational Knots, Journal of Algebra, 310 (2007), 491-525; arXiv:math/0210174 [math.GT], 2002.
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,1,0,-2).

Crossrefs

Cf. A000225, A018240, A089891, A089892, A089797, A051449, A214927, A051450, A078478.

Programs

Mathematica

CoefficientList[Series[(2 x^7 + 2 x^6 - x^5 + x^3 - x^2 + x + 1) / ((x-1) (x+1) (x^2+1) (2 x^2-1)), {x, 0, 50}], x] (* Vincenzo Librandi, May 17 2013 *)

PARI

Vec(x^3*(1+x-x^2+x^3-x^5+2*x^6+2*x^7)/((1-x)*(1+x)*(1+x^2)*(1-2*x^2)) + O(x^60)) \\ Colin Barker, Dec 26 2015

Formula

G.f.: x^3*(1+x-x^2+x^3-x^5+2*x^6+2*x^7) / ((1-x)*(1+x)*(1+x^2)*(1-2*x^2)).

a(n) = 2*a(n-2)+a(n-4)-2*a(n-6) for n>10. - Colin Barker, Dec 26 2015