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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

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OFFSET

1,3

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

Table of n, a(n) for n=1..28.

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

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

a(n) = (Lucas(2*n-1) - (if n == 2 (mod 3) then 2 else 1))/2. - Gerry Martens, Sep 06 2025

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 *)

a[n_] := (LucasL[2 n - 1] - If[Mod[n, 3] == 2, 2, 1])/2

Table[a[n], {n, 1, 30}] (* Gerry Martens, Sep 06 2025 *)

CROSSREFS

Cf. A071618, A110528, A301774.

Sequence in context: A224716 A127980 A054486 * A196976 A045553 A270687

Adjacent sequences: A072127 A072128 A072129 * A072131 A072132 A072133

KEYWORD

nonn,easy

AUTHOR

Robert G. Wilson v, Jun 24 2002

STATUS

approved

URL: https://oeis.org/A072130

⇱ A072130 - OEIS