A062882 a(n) = (1 - 2*cos(Pi/9))^n + (1 + 2*cos(Pi*2/9))^n + (1 + 2*cos(Pi*4/9))^n.

3, 9, 18, 45, 108, 270, 675, 1701, 4293, 10854, 27459, 69498, 175932, 445419, 1127763, 2855493, 7230222, 18307377, 46355652, 117376290, 297206739, 752553261, 1905530913, 4824972522, 12217257783, 30935180610, 78330624264

Offset

1,1

Comments

From L. Edson Jeffery, Apr 05 2011: (Start)

Let U be the matrix (see [Jeffery])

U = U_(9,2) =

(0 0 1 0)

(0 1 0 1)

(1 0 1 1)

(0 1 1 1).

Then a(n) = Trace(U^n).

(End)

We note that all numbers of the form a(n)*3^(-floor((n+4)/3)) are integers. - Roman Witula, Sep 29 2012

Links

Harry J. Smith, Table of n, a(n) for n = 1..200

L. Edson Jeffery, Danzer matrices

Index entries for linear recurrences with constant coefficients, signature (3,0,-3).

Formula

G.f.: x*(3 - 9*x^2)/(1 - 3*x + 3*x^3). The terms in parentheses in the definition are the roots of x^3-3*x^2+3. - Ralf Stephan, Apr 10 2004

a(n) = 3*(a(n-1) - a(n-3)) for n >= 4 - Roman Witula, Sep 29 2012

Example

We have a(2)=3*a(1), a(4)/a(3) = a(6)/a(5) = a(7)/a(6) = 5/2, a(6)=6*a(4), a(7)=15*a(4), and (1 + c(1))^8 + (1 + c(2))^8 + (1 + c(4))^8 = 7*3^5. - Roman Witula, Sep 29 2012

Maple

Digits := 1000:q := seq(floor(evalf((1-2*cos(1/9*Pi))^n+(1+2*cos(2/9*Pi))^n+(1+2*cos(4/9*Pi))^n)), n=1..50);

Mathematica

LinearRecurrence[{3, 0, -3}, {3, 9, 18}, 25] (* Georg Fischer Feb 02 2019 *)

Prog

(PARI) { default(realprecision, 200); for (n=1, 200, a=(1 - 2*cos(1/9*Pi))^n + (1 + 2*cos(2/9*Pi))^n + (1 + 2*cos(4/9*Pi))^n; write("b062882.txt", n, " ", round(a)) ) } \\ Harry J. Smith, Aug 12 2009
(PARI)  Vec((3-9*x^2)/(1-3*x+3*x^3)+O(x^66)) /* Joerg Arndt, Apr 08 2011 */

Crossrefs

Cf. A033304, A062883.

Sequence in context: A057578 A271370 A015635 * A363534 A027101 A015656

Adjacent sequences: A062879 A062880 A062881 * A062883 A062884 A062885

Keyword

nonn,easy

Author

Vladeta Jovovic, Jun 27 2001

Extensions

More terms from Sascha Kurz, Mar 24 2002

Adapted formula, denominator of g.f. and modified g.f. (and offset) to accommodate added initial term a(0)=4. - L. Edson Jeffery, Apr 05 2011

a(0) = 4 removed, g.f. and programs adapted by Georg Fischer, Feb 02 2019

Status

approved