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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
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
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
Sequence in context: A308731 A025194 A084156 * A212501 A117717 A359252
KEYWORD
nonn,easy
AUTHOR
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.)