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A020698
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a(n) = 5*a(n-1)-2*a(n-2), with a(0)=2, a(1)=9.
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3
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2, 9, 41, 187, 853, 3891, 17749, 80963, 369317, 1684659, 7684661, 35053987, 159900613, 729395091, 3327174229, 15177080963, 69231056357, 315801119859, 1440543486581, 6571115193187, 29974488992773, 136730214577491, 623702094901909, 2845050045354563
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Coincides with Pisot sequences L(2,9), E(2,9) (at least for first 1000 terms).
Number of ways to 3-color a 3 X (n+1) rectangular grid ignoring permutations of the colors. [Andrew Woods, Sep 07 2011]
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REFERENCES
| S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78).
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (5,-2).
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FORMULA
| a(k-1) = [M^k]_1,3, where M is the 3 by 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. [From Milan R. Janjic (agnus(AT)blic.net), May 08 2010]
Contribution by 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)
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MATHEMATICA
| LinearRecurrence[{5, -2}, {2, 9}, 30] (* From Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
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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
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CROSSREFS
| See A008776 for definitions of Pisot sequences
Cf. A078099.
Sequence in context: A002825 A052322 A130767 * A128752 A074611 A181375
Adjacent sequences: A020695 A020696 A020697 * A020699 A020700 A020701
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KEYWORD
| nonn,easy
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AUTHOR
| David W. Wilson (davidwwilson(AT)comcast.net)
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