A102902 a(n) = 9*a(n-1) - 16*a(n-2), with a(0) = 1, a(1) = 9.

1, 9, 65, 441, 2929, 19305, 126881, 833049, 5467345, 35877321, 235418369, 1544728185, 10135859761, 66507086889, 436390025825, 2863396842201, 18788331166609, 123280631024265, 808912380552641, 5307721328585529

Offset

0,2

Links

Indranil Ghosh, Table of n, a(n) for n = 0..1221

R. Flórez, R. A. Higuita, and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5, Journal of Integer Sequences, Vol. 17 (2014).

Index entries for linear recurrences with constant coefficients, signature (9,-16).

Formula

G.f.: 1/(1-9*x+16*x^2).

a(n) = Sum_{k=0..n} binomial(2*n-k+1, k)*4^k.

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-16)^k*9^(n-2*k).

a(n) = (1/sqrt(17))*( ((9+sqrt(17))/2)^(n+1) - ((9-sqrt(17))/2)^(n+1) ), with n >= 0. - Paolo P. Lava, Jun 16 2008

a(n) = 4^n * ChebyshevU(n, 9/8). - G. C. Greubel, Dec 09 2022

Mathematica

LinearRecurrence[{9, -16}, {1, 9}, 20] (* Harvey P. Dale, Jul 28 2016 *)

Prog

(SageMath) [lucas_number1(n, 9, 16) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009
(Magma) [4^n*Evaluate(ChebyshevSecond(n+1), 9/8): n in [0..30]]; // G. C. Greubel, Dec 09 2022

Crossrefs

Cf. A002540, A099459.

Sequence in context: A055284 A351530 A081040 * A127534 A037548 A238275

Adjacent sequences: A102899 A102900 A102901 * A102903 A102904 A102905

Keyword

easy,nonn

Author

Paul Barry, Jan 17 2005

Status

approved