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

Author

David W. Wilson

Keywords

• nonn
• easy

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).

Crossrefs

See A008776 for definitions of Pisot sequences.

Cf. A078099.

Programs

Magma

m:=24; R:=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

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 *)

PARI

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

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)

E.g.f.: 2*exp(5*x/2)*(17*cosh(sqrt(17)*x/2) + 4*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, Jun 17 2025