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A002534 a(n) = 2*a(n-1) + 9*a(n-2), with a(0) = 0, a(1) = 1.

%I A002534 M2058 N0814 #68 Jun 01 2025 05:55:36
%S A002534 0,1,2,13,44,205,806,3457,14168,59449,246410,1027861,4273412,17797573,
%T A002534 74055854,308289865,1283082416,5340773617,22229288978,92525540509,
%U A002534 385114681820,1602959228221,6671950592822,27770534239633,115588623814664
%N A002534 a(n) = 2*a(n-1) + 9*a(n-2), with a(0) = 0, a(1) = 1.
%C A002534 For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 2's along the main diagonal, and 3's along the superdiagonal and the subdiagonal. - _John M. Campbell_, Jul 19 2011
%D A002534 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002534 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A002534 A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.
%H A002534 Vincenzo Librandi, <a href="/A002534/b002534.txt">Table of n, a(n) for n = 0..1000</a>
%H A002534 J. Borowska and L. Lacinska, <a href="https://doi.org/10.17512/jamcm.2014.4.03">Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix</a>, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=2, b=3.
%H A002534 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A002534 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A002534 Albert Tarn, <a href="/A001333/a001333_1.pdf">Approximations to certain square roots and the series of numbers connected therewith</a> [Annotated scanned copy]
%H A002534 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,9).
%F A002534 From _Paul Barry_, Sep 29 2004: (Start)
%F A002534 E.g.f.: exp(x)*sinh(sqrt(10)*x)/sqrt(10).
%F A002534 a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*10^k. (End)
%F A002534 a(n) = ((1+sqrt(10))^n - (1-sqrt(10))^n)/(2*sqrt(10)). - _Artur Jasinski_, Dec 10 2006
%F A002534 G.f.: x/(1 - 2*x - 9*x^2) - _Iain Fox_, Jan 17 2018
%F A002534 From _G. C. Greubel_, Jan 03 2024: (Start)
%F A002534 a(n) = (3*i)^(n-1)*ChebyshevU(n-1, -i/3).
%F A002534 a(n) = 3^(n-1)*Fibonacci(n, 2/3), where Fibonacci(n, x) is the Fibonacci polynomial. (End)
%p A002534 A002534:=-z/(-1+2*z+9*z**2); # [_Simon Plouffe_ in his 1992 dissertation.]
%t A002534 LinearRecurrence[{2, 9}, {0, 1}, 30] (* _T. D. Noe_, Aug 18 2011 *)
%o A002534 (Sage) [lucas_number1(n,2,-9) for n in range(0, 20)] # _Zerinvary Lajos_, Apr 22 2009
%o A002534 (Magma) [Ceiling(((1+Sqrt(10))^n-(1-Sqrt(10))^n)/(2*Sqrt(10))): n in [0..30]]; // _Vincenzo Librandi_, Aug 15 2011
%o A002534 (PARI) first(n) = Vec(x/(1 - 2*x - 9*x^2) + O(x^n), -n) \\ _Iain Fox_, Jan 17 2018
%Y A002534 Cf. A015445, A099012.
%K A002534 nonn,easy
%O A002534 0,3
%A A002534 _N. J. A. Sloane_
%E A002534 More terms from _Johannes W. Meijer_, Aug 18 2011