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A221719
a(n) = 3*2^n - Fibonacci(n+3) - 1.
2
0, 2, 6, 15, 34, 74, 157, 328, 678, 1391, 2838, 5766, 11677, 23588, 47554, 95719, 192426, 386450, 775485, 1555152, 3117070, 6245087, 12507886, 25044430, 50135229, 100345484, 200812362, 401821143, 803960098, 1608434426, 3217700893, 6436748056, 12875674422, 25754873423, 51515449734, 103040126934, 206095184221, 412214526260, 824468140690
OFFSET
0,2
COMMENTS
Number of 3-strand braids with n crossings, divided by 2.
REFERENCES
Paul K. Stockmeyer, Personal communication, Jan 12 2013
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
Cf. A000045.
Sequence in context: A101352 A192980 A281970 * A095380 A287012 A355061
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 31 2013
STATUS
approved