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

%I #49 May 13 2024 17:28:50

%S 1,7,38,189,905,4256,19837,92043,426094,1970185,9104261,42057792,

%T 194257673,897167999,4143341590,19134543141,88365044497,408075336928,

%U 1884511869029,8702754376995,40189650079646,185597252410577,857094615997933,3958092535469184,18278606972307025

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

%C 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.

%H G. C. Greubel, <a href="/A099453/b099453.txt">Table of n, a(n) for n = 0..1000</a>

%H Dror Bar-Natan, <a href="http://katlas.org/wiki/Main_Page">The Rolfsen Knot Table</a>.

%H S. Falcon, <a href="http://dx.doi.org/10.9734/BJMCS/2014/11783">Iterated Binomial Transforms of the k-Fibonacci Sequence</a>, British Journal of Mathematics & Computer Science, 4 (22): 2014.

%H J. Pan, <a href="https://cs.uwaterloo.ca/journals/JIS/OL13/Pan/pan8.html">Multiple Binomial Transforms and Families of Integer Sequences </a>, J. Int. Seq. 13 (2010), 10.4.2, F^(3).

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-11).

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

%F 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

%F a(n) = 7*a(n-1) - 11*a(n-2), n >= 2. - _Vincenzo Librandi_, Mar 18 2011

%F 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

%t LinearRecurrence[{7,-11}, {1,7}, 30] (* _G. C. Greubel_, May 21 2019 *)

%o (Sage) [lucas_number1(n,7,11) for n in range(1, 30)] # _Zerinvary Lajos_, Apr 23 2009

%o (PARI) Vec(1/(1-7*x+11*x^2) + O(x^30)) \\ _Michel Marcus_, Sep 09 2017

%o (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

%o (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

%Y Cf. A030191, A099454, A099455.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Oct 16 2004