cp's OEIS Frontend

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

%I A078477 #29 Jun 09 2019 18:42:39
%S A078477 1,1,1,3,3,6,7,15,15,30,31,63,63,126,127,255,255,510,511,1023,1023,
%T A078477 2046,2047,4095,4095,8190,8191,16383,16383,32766,32767,65535,65535,
%U A078477 131070,131071,262143,262143,524286,524287,1048575,1048575,2097150,2097151
%N A078477 Number of rational knots with n crossings and unknotting number = 1 (chiral pairs counted only once).
%C A078477 From _Alexander Adamchuk_, Nov 16 2009: (Start)
%C A078477 For n>1 a(2n+1) = 2^(n-1) - 1 = A000225(n-1).
%C A078477 For n>1 a(4n) = a(4n+1) - 1 = 2^(2n-1) - 2.
%C A078477 For n>0 a(4n+2) = a(4n+3) = 2^(2n) - 1. (End)
%H A078477 Andrey Zabolotskiy, <a href="/A078477/b078477.txt">Table of n, a(n) for n = 3..1000</a>
%H A078477 A. Stoimenow, <a href="https://doi.org/10.1016/j.jalgebra.2006.11.031">Generating Functions, Fibonacci Numbers and Rational Knots</a>, Journal of Algebra, 310 (2007), 491-525; arXiv:<a href="https://arxiv.org/abs/math/0210174">math/0210174 [math.GT]</a>, 2002.
%H A078477 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,1,0,-2).
%F A078477 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)).
%F A078477 a(n) = 2*a(n-2)+a(n-4)-2*a(n-6) for n>10. - _Colin Barker_, Dec 26 2015
%t A078477 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 *)
%o A078477 (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
%Y A078477 Cf. A000225, A018240, A089891, A089892, A089797, A051449, A214927, A051450, A078478.
%K A078477 nonn,easy
%O A078477 3,4
%A A078477 _Ralf Stephan_, Jan 03 2003