This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A020698 #65 Jun 17 2025 16:03:13
%S A020698 2,9,41,187,853,3891,17749,80963,369317,1684659,7684661,35053987,
%T A020698 159900613,729395091,3327174229,15177080963,69231056357,315801119859,
%U A020698 1440543486581,6571115193187,29974488992773,136730214577491,623702094901909,2845050045354563
%N A020698 a(n) = 5*a(n-1) - 2*a(n-2), with a(0)=2, a(1)=9.
%C A020698 Coincides with Pisot sequence L(2,9) (at least for first 1000 terms).
%C A020698 Coincides with Pisot sequence E(2,9) (at least for first 1000 terms).
%C A020698 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 A020698 Number of ways to 3-color a 3 X (n+1) rectangular grid ignoring permutations of the colors. - _Andrew Woods_, Sep 07 2011
%D A020698 S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78).
%H A020698 Vincenzo Librandi, <a href="/A020698/b020698.txt">Table of n, a(n) for n = 0..1000</a>
%H A020698 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 A020698 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-2).
%F A020698 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 A020698 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 A020698 From _Bruno Berselli_, Sep 06 2011: (Start)
%F A020698 G.f.: (2-x)/(1-5*x+2*x^2).
%F A020698 a(n) = ((17+4*sqrt(17))*(5+sqrt(17))^n+(17-4*sqrt(17))*(5-sqrt(17))^n)/(17*2^n).
%F A020698 a(-n)*2^n = A052984(n-2). (End)
%F A020698 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
%t A020698 LinearRecurrence[{5,-2},{2,9},30] (* _Vladimir Joseph Stephan Orlovsky_, Jan 29 2012 *)
%t A020698 CoefficientList[Series[(2 - x)/(1 - 5 x + 2 x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Mar 19 2013 *)
%o A020698 (PARI) a(n)=([2,1,2;1,1,1;2,1,2]^(n+1))[1,3]
%o A020698 (Magma) m:=24; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((2-x)/(1-5*x+2*x^2))); // _Bruno Berselli_, Sep 06 2011
%o A020698 (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 A020698 See A008776 for definitions of Pisot sequences.
%Y A020698 Cf. A078099.
%K A020698 nonn,easy
%O A020698 0,1
%A A020698 _David W. Wilson_