A221719 a(n) = 3*2^n - Fibonacci(n+3) - 1.

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

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

Cf. A000045.

Sequence in context: A101352 A192980 A281970 * A095380 A287012 A355061

Adjacent sequences: A221716 A221717 A221718 * A221720 A221721 A221722

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 31 2013

Status

approved