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

Offset: 0

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• nonn

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

Crossrefs

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

Programs

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

Mathematica

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

Sage

[lucas_number1(n,7,9) for n in range(1, 22)] # Zerinvary Lajos, Apr 23 2009

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)