A072130 a(n+1) - 3*a(n) + a(n-1) = (2/3)*(1+w^(n+1)+w^(2*n+2)); a(1) = 0, a(2) = 1; where w is the cubic root of unity.

0, 1, 5, 14, 37, 99, 260, 681, 1785, 4674, 12237, 32039, 83880, 219601, 574925, 1505174, 3940597, 10316619, 27009260, 70711161, 185124225, 484661514, 1268860317, 3321919439, 8696898000, 22768774561, 59609425685, 156059502494

Offset: 1

Author

Robert G. Wilson v, Jun 24 2002

Keywords

• nonn
• easy

Comments

w = exp(2*Pi*I/3) = (-1-sqrt(-3))/2.

The sequence (2/3)*(1+w^(n+1)+w^(2*n+2)) is "Period 3: repeat [0,2,0]."

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-1,1,-3,1).

Crossrefs

Cf. A071618.

Programs

Mathematica

a[1] = 0; a[2] = 1; w = Exp[2Pi*I/3]; a[n_] := (2/3)(1 + w^n + w^(2n)) + 3a[n - 1] - a[n - 2]; Table[ Simplify[ a[n]], {n, 1, 28}]
LinearRecurrence[{3,-1,1,-3,1},{0,1,5,14,37},30] (* Harvey P. Dale, Aug 19 2012 *)

Formula

G.f.: x^2*(1+x)*(1+x-x^2)/((1-x)*(1-3*x+x^2)*(1+x+x^2)). - Colin Barker, Jan 14 2012

a(n) = 3*a(n-1)- a(n-2)+ a(n-3)-3*a(n-4)+a(n-5). - Harvey P. Dale, Aug 19 2012