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

Author

N. J. A. Sloane, Jan 31 2013

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000045, A083329, A104004.

Programs

Magma

A221719:= func< n | 3*2^n - Fibonacci(n+3) - 1 >; // G. C. Greubel, Jun 05 2025

Mathematica

LinearRecurrence[{4,-4,-1,2},{0,2,6,15},40] (* Harvey P. Dale, Aug 25 2015 *)
A221719[n_]:= 3*2^n -Fibonacci[n+3] -1; (* G. C. Greubel, Jun 05 2025 *)

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

SageMath

def A221719(n): return 3*2**n - fibonacci(n+3) - 1 # G. C. Greubel, Jun 05 2025

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*(2-2*x-x^2) / ((1-x)*(1-2*x)*(1-x-x^2)). (End)

a(n) = -1 + 3*2^n + ( (2-sqrt(5))*((1-sqrt(5))/2)^n - (2+sqrt(5))*((1+sqrt(5))/2)^n )/sqrt(5). - Colin Barker, Nov 03 2016

From G. C. Greubel, Jun 05 2025: (Start)

a(n) = A083329(n+1) - A000045(n+3).

a(n) = A104004(n) - 1.

E.g.f.: 3*exp(2*x) - exp(x) - (2/sqrt(5))*exp(x/2)*( 2*sinh(sqrt(5)*x/2) + sqrt(5)*cosh(sqrt(5)*x/2) ). (End)