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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: A103218 A319311 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 January 22 23:00 EST 2019. Contains 319365 sequences. (Running on oeis4.)