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A020698
a(n) = 5*a(n-1) - 2*a(n-2), with a(0)=2, a(1)=9.
7
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
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)
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
KEYWORD
nonn,easy
STATUS
approved