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A099453 Expansion of 1/(1 - 7*x + 11*x^2). 11
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 (list; graph; refs; listen; history; text; internal format)
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
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).
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
Sequence in context: A296769 A241524 A291822 * A292535 A026763 A217340
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 16 2004
STATUS
approved

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Last modified June 14 05:17 EDT 2024. Contains 373393 sequences. (Running on oeis4.)