OFFSET
0,2
COMMENTS
Number of 3-strand braids with n crossings, divided by 2.
REFERENCES
Paul K. Stockmeyer, Personal communication, Jan 12 2013
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,-1,2).
FORMULA
From Colin Barker, Jun 07 2015: (Start)
a(n) = 4*a(n-1)-4*a(n-2)-a(n-3)+2*a(n-4) for n>3.
G.f.: x*(x^2+2*x-2) / ((x-1)*(2*x-1)*(x^2+x-1)). (End)
a(n) = (-1+3*2^n-(2^(-n)*((1-sqrt(5))^n*(-2+sqrt(5))+(1+sqrt(5))^n*(2+sqrt(5))))/sqrt(5)). - Colin Barker, Nov 03 2016
MATHEMATICA
LinearRecurrence[{4, -4, -1, 2}, {0, 2, 6, 15}, 40] (* Harvey P. Dale, Aug 25 2015 *)
PROG
(PARI) concat(0, Vec(x*(x^2+2*x-2)/((x-1)*(2*x-1)*(x^2+x-1)) + O(x^100))) \\ Colin Barker, Jun 07 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 31 2013
STATUS
approved