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%I #41 Sep 08 2022 08:45:15
%S 1,7,40,217,1159,6160,32689,173383,919480,4875913,25856071,137109280,
%T 727060321,3855438727,20444528200,108412748857,574888488199,
%U 3048504677680,16165536349969,85722212350663,454565659304920
%N Expansion of 1/(1 - 7*x + 9*x^2).
%C 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.
%H G. C. Greubel, <a href="/A099459/b099459.txt">Table of n, a(n) for n = 0..1000</a>
%H Dror Bar-Natan, <a href="http://katlas.org/wiki/9_48">9 48</a>, The Knot Atlas.
%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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-9).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-9)^k*7^(n-2*k).
%F a(n) = Sum{k=0..n} binomial(2*n-k+1, k) * 3^k. - _Paul Barry_, Jan 17 2005
%F a(n) = 7*a(n-1) - 9*a(n-2), n >= 2. - _Vincenzo Librandi_, Mar 18 2011
%F a(n) = ((7 + sqrt(13))^(n+1) - (7 - sqrt(13))^(n+1))/(2^(n+1)*sqrt(13)). - _Rolf Pleisch_, May 19 2011
%F a(n) = 3^(n-1)*ChebyshevU(n-1, 7/6). - _G. C. Greubel_, Nov 18 2021
%t LinearRecurrence[{7,-9},{1,7},30] (* _Harvey P. Dale_, Jan 06 2012 *)
%o (Sage) [lucas_number1(n,7,9) for n in range(1, 22)] # _Zerinvary Lajos_, Apr 23 2009
%o (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
%Y Cf. A002540, A099460, A099461.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Oct 16 2004