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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A015445, A099012.

Programs

Magma

[Ceiling(((1+Sqrt(10))^n-(1-Sqrt(10))^n)/(2*Sqrt(10))): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

Maple

A002534:=-z/(-1+2*z+9*z**2); # [Simon Plouffe in his 1992 dissertation.]

Mathematica

LinearRecurrence[{2, 9}, {0, 1}, 30] (* T. D. Noe, Aug 18 2011 *)

PARI

first(n) = Vec(x/(1 - 2*x - 9*x^2) + O(x^n), -n) \\ Iain Fox, Jan 17 2018

Sage

[lucas_number1(n,2,-9) for n in range(0, 20)] # Zerinvary Lajos, Apr 22 2009

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)

Extensions

More terms from Johannes W. Meijer, Aug 18 2011