A099459 Expansion of 1/(1 - 7*x + 9*x^2).

1, 7, 40, 217, 1159, 6160, 32689, 173383, 919480, 4875913, 25856071, 137109280, 727060321, 3855438727, 20444528200, 108412748857, 574888488199, 3048504677680, 16165536349969, 85722212350663, 454565659304920, 2410459703978473, 12782126994105031, 67780751622928960

Offset

0,2

Comments

Associated to the knot 9_48 by the modified Chebyshev transform A(x) -> (1/(1+x^2)^2)*A(x/(1+x^2)). See A099460 and A099461.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Dror Bar-Natan, 9 48, The Knot Atlas.

Sergio Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22) (2014), 3135-3145.

Index entries for linear recurrences with constant coefficients, signature (7,-9).

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-9)^k*7^(n-2*k).

a(n) = Sum_{k=0..n} binomial(2*n-k+1, k) * 3^k. - Paul Barry, Jan 17 2005

a(n) = 7*a(n-1) - 9*a(n-2), n >= 2. - Vincenzo Librandi, Mar 18 2011

a(n) = ((7 + sqrt(13))^(n+1) - (7 - sqrt(13))^(n+1))/(2^(n+1)*sqrt(13)). - Rolf Pleisch, May 19 2011

a(n) = 3^(n-1)*ChebyshevU(n-1, 7/6). - G. C. Greubel, Nov 18 2021

From Peter Bala, Jul 23 2025: (Start)

The following products telescope:

Product_{k >= 1} 1 + 3^k/a(k) = (1 + sqrt(13))/2.

Product_{k >= 1} 1 - 3^k/a(k) = (1 + sqrt(13))/14,

Product_{k >= 1} 1 + (-3)^k/a(k) = (13 + sqrt(13))/26.

Product_{k >= 1} 1 - (-3)^k/a(k) = (13 + sqrt(13))/14. (End)

Sum_{n >= 1} a(n-1)*x^(2*n)/(2*n)! = (2/sqrt(13))*sinh(sqrt(13)*x/2)*sinh(x/2). - Peter Bala, Sep 22 2025

E.g.f.: exp(7*x/2) * (cosh(sqrt(13)*x/2) + 7*sinh(sqrt(13)*x/2)/sqrt(13)). - Amiram Eldar, Feb 12 2026

Mathematica

LinearRecurrence[{7, -9}, {1, 7}, 30] (* Harvey P. Dale, Jan 06 2012 *)

Prog

(SageMath) [lucas_number1(n, 7, 9) for n in range(1, 22)] # Zerinvary Lajos, Apr 23 2009
(Magma) [n le 2 select 7^(n-1) else 7*Self(n-1) -9*Self(n-2): n in [1..31]]; // G. C. Greubel, Nov 18 2021

Crossrefs

Cf. A002450, A004187, A016153, A099460, A099461, A102902.

Sequence in context: A083327 A026716 A026782 * A243875 A360101 A371813

Adjacent sequences: A099456 A099457 A099458 * A099460 A099461 A099462

Keyword

easy,nonn

Author

Paul Barry, Oct 16 2004

Status

approved