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

3,4

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).

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A050065 A367394 A298732 * A336077 A360375 A098832

Adjacent sequences: A078474 A078475 A078476 * A078478 A078479 A078480

Keyword

nonn,easy

Author

Ralf Stephan, Jan 03 2003

Status

approved