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

Author

Vladeta Jovovic, Jun 27 2001

Keywords

• nonn
• easy

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

Examples

			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
		

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).

Crossrefs

Cf. A033304, A062883.

Programs

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 *)

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 */

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

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