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%I #56 Jan 02 2024 09:33:58
%S 1,2,-5,-28,-11,230,559,-952,-6935,-5302,51811,151340,-163619,
%T -1689298,-1906025,11391632,39937489,-22649710,-404736821,-605626252,
%U 2431378885,10313394038,-1255621889
%N Expansion of 1/(1 - 2*x + 9*x^2).
%C Hankel transform of A100193. A member of the family of sequences with g.f. 1/(1-2*x+r^2*x^2) which are the Hankel transforms of the sequences given by Sum_{k=0..n} binomial(2*n,k)*r^(n-k).
%C From _Peter Bala_, Apr 01 2018: (Start)
%C With offset 1, this is the Lucas sequence U(n,2,9). The companion Lucas sequence V(n,2,9) is 2*A025172(n).
%C Define a binary operation o on rational numbers by x o y = (x + y)/(1 - 2*x*y). This is a commutative and associative operation with identity 0. Then 2 o 2 o ... o 2 (n terms) = 2*A127357(n-1)/A025172(n). Cf. A088137 and A087455. (End)
%H G. C. Greubel, <a href="/A127357/b127357.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lucas_sequence">Lucas sequence</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-9).
%F a(n) = Sum_{k=0..n} binomial(n-k,k)*2^(n-2*k)*(-9)^k.
%F a(n) = 2*a(n-1) - 9*a(n-2) for n >= 2. - _Vincenzo Librandi_, Mar 22 2011
%F a(n) = ((1-2*sqrt(2)*i)^n-(1+2*sqrt(2)*i)^n)*i/(4*sqrt(2)), where i=sqrt(-1). - _Bruno Berselli_, Jul 01 2011
%F From _Vladimir Reshetnikov_, Oct 15 2016: (Start)
%F a(n) = 3^n*(cos(n*theta) + sin(n*theta)*sqrt(2)/4), theta = arctan(2*sqrt(2)).
%F E.g.f.: exp(x)*(cos(2*sqrt(2)*x) + sin(2*sqrt(2)*x)*sqrt(2)/4). (End)
%F a(n) = 2^n*Product_{k=1..n}(1 + 3*cos(k*Pi/(n+1))). - _Peter Luschny_, Nov 28 2019
%F From _G. C. Greubel_, Jan 02 2024: (Start)
%F a(n) = (-1)^n * A025170(n).
%F a(n) = 3^n * ChebyshevU(n, 1/3). (End)
%p c := 2*sqrt(2): g := exp(x)*(sin(c*x)+c*cos(c*x))/c: ser := series(g,x,32):
%p seq(n!*coeff(ser,x,n), n=0..22); # _Peter Luschny_, Oct 19 2016
%t RootReduce@Table[3^n (Cos[n ArcTan[2 Sqrt[2]]] + Sin[n ArcTan[2 Sqrt[2]]] Sqrt[2]/4), {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 15 2016 *)
%t CoefficientList[Series[1/(1-2x+9x^2),{x,0,40}],x] (* or *)
%t LinearRecurrence[ {2,-9},{1,2},40] (* _Harvey P. Dale_, Mar 15 2022 *)
%t Table[3^n*ChebyshevU[n, 1/3], {n,0,40}] (* _G. C. Greubel_, Jan 02 2024 *)
%o (Sage) [lucas_number1(n,2,9) for n in range(1, 24)] # _Zerinvary Lajos_, Apr 23 2009
%o (SageMath) [3^n*chebyshev_U(n,1/3) for n in range(41)] # _G. C. Greubel_, Jan 02 2024
%o (Magma) m:=23; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-2*x+9*x^2))); // _Bruno Berselli_, Jul 01 2011
%o (Magma) [3^n*Evaluate(ChebyshevU(n+1),1/3): n in [0..50]]; // _G. C. Greubel_, Jan 02 2024
%o (Maxima) makelist(coeff(taylor(1/(1-2*x+9*x^2), x, 0, n), x, n), n, 0, 22); /* _Bruno Berselli_, Jul 01 2011 */
%o (PARI) Vec(1/(1-2*x+9*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o (GAP) a:=[1,2];; for n in [3..25] do a[n]:=2*a[n-1]-9*a[n-2]; od; a; # _Muniru A Asiru_, Oct 23 2018
%Y Variant is A025170.
%Y Cf. A025172, A087455, A088137, A100193, A127357.
%K sign,easy
%O 0,2
%A _Paul Barry_, Jan 11 2007
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