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

2, 9, 41, 187, 853, 3891, 17749, 80963, 369317, 1684659, 7684661, 35053987, 159900613, 729395091, 3327174229, 15177080963, 69231056357, 315801119859, 1440543486581, 6571115193187, 29974488992773, 136730214577491, 623702094901909, 2845050045354563

Offset

0,1

Comments

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

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

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

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

References

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

S. B. Ekhad, N. J. A. Sloane, D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT] (2016)

Index entries for linear recurrences with constant coefficients, signature (5,-2).

Formula

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

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

From Bruno Berselli, Sep 06 2011: (Start)

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

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

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

Mathematica

LinearRecurrence[{5, -2}, {2, 9}, 30] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
CoefficientList[Series[(2 - x)/(1 - 5 x + 2 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 19 2013 *)

Prog

(PARI) a(n)=([2, 1, 2; 1, 1, 1; 2, 1, 2]^(n+1))[1, 3]
(Magma) m:=24; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((2-x)/(1-5*x+2*x^2))); // Bruno Berselli, Sep 06 2011
(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

Crossrefs

See A008776 for definitions of Pisot sequences.

Cf. A078099.

Sequence in context: A130767 A273461 A217190 * A128752 A074611 A362381

Adjacent sequences: A020695 A020696 A020697 * A020699 A020700 A020701

Keyword

nonn,easy

Author

David W. Wilson

Status

approved