A002534 a(n) = 2*a(n-1) + 9*a(n-2), with a(0) = 0, a(1) = 1. (Formerly M2058 N0814)

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

0,3

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