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%I #47 Dec 30 2023 23:50:34
%S 0,1,9,72,567,4455,34992,274833,2158569,16953624,133155495,1045816839,
%T 8213952096,64513217313,506693386953,3979621526760,31256353258263,
%U 245490585583527,1928108090927376,15143557548094641,118939045114505385,934159388097696696
%N a(n) = 9*a(n-1) - 9*a(n-2) for n>1, with a(0)=0, a(1)=1.
%C Scaled Chebyshev U-polynomials evaluated at 3/2.
%H Colin Barker, <a href="/A057085/b057085.txt">Table of n, a(n) for n = 0..1000</a>
%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=9, q=-9.
%H Wolfdieter Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419. Eqs.(38) and (45),lhs, m=9.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-9).
%F a(n) = A001906(n)*3^(n-1).
%F a(n) = S(n, 3)*3^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
%F a(n) = A001906(n)*A000244(n)/3. - _Robert G. Wilson v_, Sep 21 2006
%F a(2k) = A004187(k)*9^k/3, a(2k-1) = A033890(k)*9^k.
%F G.f.: x/(1-9*x+9*x^2).
%F a(n) = (1/3)*Sum_{k=0..n} binomial(n, k)*Fibonacci(4*k). - _Benoit Cloitre_, Jun 21 2003
%F a(n+1) = Sum_{k=0..n} A109466(n,k)*9^k. - _Philippe Deléham_, Oct 28 2008
%t f[n_]:= Fibonacci[2n]*3^(n-1); Table[f@n, {n, 0, 20}] (* or *)
%t a[0]=0; a[1]=1; a[n_]:= a[n]= 9(a[n-1] -a[n-2]); Table[a[n], {n, 0, 20}] (* or *)
%t CoefficientList[Series[x/(1-9x +9x^2), {x, 0, 20}], x] (* _Robert G. Wilson v_ Sep 21 2006 *)
%o (PARI) a(n)=(1/3)*sum(k=0,n,binomial(n,k)*fibonacci(4*k)) \\ _Benoit Cloitre_
%o (PARI) concat(0, Vec(x/(1-9*x+9*x^2) + O(x^30))) \\ _Colin Barker_, Jun 14 2015
%o (Sage) [lucas_number1(n,9,9) for n in range(0, 21)] # _Zerinvary Lajos_, Apr 23 2009
%o (Magma) [3^(n-1)*Fibonacci(2*n): n in [0..30]]; // _G. C. Greubel_, May 02 2022
%Y Cf. A000045, A000244, A001906, A004187, A030191, A033890, A049310, A109466.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Aug 11 2000
%E Edited by _N. J. A. Sloane_, Sep 16 2005