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A062882 a(n+1)=(1-2*cos(1/9*Pi))^n+(1+2*cos(2/9*Pi))^n+(1+2*cos(4/9*Pi))^n. 2
4, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

Let U be the unit-primitive 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. E. Jeffery, Unit-primitive matrices

FORMULA

G.f.: (4-9*x+3*x^3)/(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)), n=4,5,..., (if we "correct" the definition of the sequence a(n) as follows a(n) = ()^n + ()^n + ()^n and we put a(0) = 3 instead of a(0) = 4 this recurrence relation will holds also for n=3, the basis of given a(0) = 4 by L. Edson Jeffery was the definition of power of zero order of matrix U). - 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);

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((4-9*x+3*x^3)/(1-3*x+3*x^3)+O(x^66))  /* Joerg Arndt, Apr 08 2011 */

CROSSREFS

Cf. A033304, A062883.

Sequence in context: A187770 A103218 A107381 * A242531 A275160 A132192

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

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

STATUS

approved

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Last modified February 17 22:32 EST 2018. Contains 299297 sequences. (Running on oeis4.)