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 A002534 a(n) = 2*a(n-1) + 9*a(n-2), with a(0) = 0, a(1) = 1. (Formerly M2058 N0814) 21
 0, 1, 2, 13, 44, 205, 806, 3457, 14168, 59449, 246410, 1027861, 4273412, 17797573, 74055854, 308289865, 1283082416, 5340773617, 22229288978, 92525540509, 385114681820, 1602959228221, 6671950592822, 27770534239633, 115588623814664 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). 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. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 J. Borowska and L. Lacinska, Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=2, b=3. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy] Index entries for linear recurrences with constant coefficients, signature (2,9). FORMULA From Paul Barry, Sep 29 2004: (Start) E.g.f.: exp(x)*sinh(sqrt(10)*x)/sqrt(10). a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*10^k. (End) a(n) = ((1+sqrt(10))^n - (1-sqrt(10))^n)/(2*sqrt(10)). - Artur Jasinski, Dec 10 2006 G.f.: x/(1 - 2*x - 9*x^2) - Iain Fox, Jan 17 2018 From G. C. Greubel, Jan 03 2024: (Start) a(n) = (3*i)^(n-1)*ChebyshevU(n-1, -i/3). a(n) = 3^(n-1)*Fibonacci(n, 2/3), where Fibonacci(n, x) is the Fibonacci polynomial. (End) MAPLE A002534:=-z/(-1+2*z+9*z**2); # [Simon Plouffe in his 1992 dissertation.] MATHEMATICA Table[((1 + Sqrt[10])^n - (1 - Sqrt[10])^n)/(2 Sqrt[10]), {n, 0, 30}]] (* Artur Jasinski, Dec 10 2006 *) LinearRecurrence[{2, 9}, {0, 1}, 30] (* T. D. Noe, Aug 18 2011 *) PROG (Sage) [lucas_number1(n, 2, -9) for n in range(0, 20)] # Zerinvary Lajos, Apr 22 2009 (Magma) [Ceiling(((1+Sqrt(10))^n-(1-Sqrt(10))^n)/(2*Sqrt(10))): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011 (PARI) first(n) = Vec(x/(1 - 2*x - 9*x^2) + O(x^n), -n) \\ Iain Fox, Jan 17 2018 CROSSREFS Cf. A015445, A099012. Sequence in context: A308731 A025194 A084156 * A212501 A117717 A359252 Adjacent sequences: A002531 A002532 A002533 * A002535 A002536 A002537 KEYWORD nonn,easy AUTHOR N. J. A. Sloane EXTENSIONS More terms from Johannes W. Meijer, Aug 18 2011 STATUS approved

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Last modified February 20 20:53 EST 2024. Contains 370217 sequences. (Running on oeis4.)