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 A020698 a(n) = 5*a(n-1) - 2*a(n-2), with a(0)=2, a(1)=9. 6

%I

%S 2,9,41,187,853,3891,17749,80963,369317,1684659,7684661,35053987,

%T 159900613,729395091,3327174229,15177080963,69231056357,315801119859,

%U 1440543486581,6571115193187,29974488992773,136730214577491,623702094901909,2845050045354563

%N a(n) = 5*a(n-1) - 2*a(n-2), with a(0)=2, a(1)=9.

%C Coincides with Pisot sequence L(2,9) (at least for first 1000 terms).

%C Coincides with Pisot sequence E(2,9) (at least for first 1000 terms).

%C Theorem: E(2,9) satisfies a(n) = 5 a(n - 1) 2 2 a(n - 2) for n>=2. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the conjecture. - _N. J. A. Sloane_, Sep 09 2016

%C Number of ways to 3-color a 3 X (n+1) rectangular grid ignoring permutations of the colors. - _Andrew Woods_, Sep 07 2011

%D S. J. Cyvin and I. Gutman, KekulĂ© structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78).

%H Vincenzo Librandi, <a href="/A020698/b020698.txt">Table of n, a(n) for n = 0..1000</a>

%H S. B. Ekhad, N. J. A. Sloane, D. Zeilberger, <a href="http://arxiv.org/abs/1609.05570">Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences</a>, arXiv:1609.05570 [math.NT] (2016)

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-2).

%F a(k-1) = [M^k]_1,3, where M is the 3 X 3 matrix [2,1,2; 1,1,1; 2,1,2]. - _Simone Severini_, Jun 12 2006

%F If p[i]=Fibonacci(2i+1) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - _Milan Janjic_, May 08 2010

%F From _Bruno Berselli_, Sep 06 2011: (Start)

%F G.f.: (2-x)/(1-5*x+2*x^2).

%F a(n) = ((17+4*sqrt(17))*(5+sqrt(17))^n+(17-4*sqrt(17))*(5-sqrt(17))^n)/(17*2^n).

%F a(-n)*2^n = A052984(n-2). (End)

%t LinearRecurrence[{5,-2},{2,9},30] (* _Vladimir Joseph Stephan Orlovsky_, Jan 29 2012 *)

%t CoefficientList[Series[(2 - x)/(1 - 5 x + 2 x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Mar 19 2013 *)

%o (PARI) a(n)=([2,1,2;1,1,1;2,1,2]^(n+1))[1,3]

%o (MAGMA) m:=24; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((2-x)/(1-5*x+2*x^2))); // _Bruno Berselli_, Sep 06 2011

%o (MAGMA) I:=[2, 9]; [n le 2 select I[n] else 5*Self(n-1)-2*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Mar 19 2013

%Y See A008776 for definitions of Pisot sequences.

%Y Cf. A078099.

%K nonn,easy

%O 0,1

%A _David W. Wilson_

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Last modified August 7 09:16 EDT 2020. Contains 336274 sequences. (Running on oeis4.)