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, 34826893868427505, 228518503558479081, 1499436230131471649

Offset

0,2

Links

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

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

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

From Peter Bala, Jul 23 2025: (Start)

a(n) := ((9 + sqrt(17))^(n+1) - (9 - sqrt(17))^(n+1))/(2^(n+1)*sqrt(17)).

The following products telescope:

Product_{k >= 1} 1 + 4^k/a(k) = (1 + sqrt(17))/2.

Product_{k >= 1} 1 - 4^k/a(k) = (1 + sqrt(17))/18.

Product_{k >= 1} 1 + (-4)^k/a(k) = (17 + sqrt(17))/34.

Product_{k >= 1} 1 - (-4)^k/a(k) = (17 + sqrt(17))/18. (End)

Sum_{n >= 1} a(n-1)*x^(2*n)/(2*n)! = (2/sqrt(17))*sinh(sqrt(17)*x/2)*sinh(x/2). - Peter Bala, Sep 22 2025

E.g.f.: exp(9*x/2) * (cosh(sqrt(17)*x/2) + 9*sinh(sqrt(17)*x/2)/sqrt(17)). - Amiram Eldar, Feb 10 2026

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. A002450, A004187, A016153, 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