A099453 Expansion of 1/(1 - 7*x + 11*x^2).

1, 7, 38, 189, 905, 4256, 19837, 92043, 426094, 1970185, 9104261, 42057792, 194257673, 897167999, 4143341590, 19134543141, 88365044497, 408075336928, 1884511869029, 8702754376995, 40189650079646, 185597252410577, 857094615997933, 3958092535469184, 18278606972307025

Offset

0,2

Comments

Associated to the knot 8_12 by the modified Chebyshev transform A(x)-> (1/(1+x^2)^2)*A(x/(1+x^2)). See A099454 and A099455.

Links

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

Dror Bar-Natan, The Rolfsen Knot Table.

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

J. Pan, Multiple Binomial Transforms and Families of Integer Sequences , J. Int. Seq. 13 (2010), 10.4.2, F^(3).

Index entries for linear recurrences with constant coefficients, signature (7,-11).

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-11)^k*7^(n-2k).

a(n) = ((7+sqrt(5))^n - (7-sqrt(5))^n)/(2^n*sqrt(5)), n > 0. Binomial transform of A030191 (Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2); 3rd binomial transform of Fibonacci(n). - Creighton Dement, Apr 19 2005

a(n) = 7*a(n-1) - 11*a(n-2), n >= 2. - Vincenzo Librandi, Mar 18 2011

E.g.f.: exp(7*x/2)*(5*cosh(sqrt(5)*x/2) + 7*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, May 13 2024

Mathematica

LinearRecurrence[{7, -11}, {1, 7}, 30] (* G. C. Greubel, May 21 2019 *)

Prog

(Sage) [lucas_number1(n, 7, 11) for n in range(1, 30)] # Zerinvary Lajos, Apr 23 2009
(PARI) Vec(1/(1-7*x+11*x^2) + O(x^30)) \\ Michel Marcus, Sep 09 2017
(Magma) I:=[1, 7]; [n le 2 select I[n] else 7*Self(n-1) -11*Self(n-2): n in [1..30]]; // G. C. Greubel, May 21 2019
(GAP) a:=[1, 7];; for n in [3..30] do a[n]:=7*a[n-1]-11*a[n-2]; od; a; # G. C. Greubel, May 21 2019

Crossrefs

Cf. A030191, A099454, A099455.

Sequence in context: A296769 A241524 A291822 * A292535 A026763 A217340

Adjacent sequences: A099450 A099451 A099452 * A099454 A099455 A099456

Keyword

easy,nonn

Author

Paul Barry, Oct 16 2004

Status

approved