Search: id:A057085
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%I A057085
%S A057085 0,1,9,72,567,4455,34992,274833,2158569,16953624,133155495,1045816839,
%T A057085 8213952096,64513217313,506693386953,3979621526760,31256353258263,
%U A057085 245490585583527,1928108090927376,15143557548094641,118939045114505385
%N A057085 a(0)=0, a(1)=1; for n>1, a(n)=9a(n-1)-9a(n-2).
%C A057085 Scaled Chebyshev U-polynomials evaluated at 3/2.
%D A057085 A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. 
               Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=9, q=-9.
%D A057085 W. Lang, On polynomials related to powers of the generating function 
               of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs.(38) and 
               (45),lhs, m=9.
%H A057085 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057085 Also Fibonacci(2n)*3^(n-1).
%F A057085 a(n) = S(n, 3)*3^n with S(n, x) := U(n, x/2), Chebyshev's polynomials 
               of the 2nd kind, A049310.
%F A057085 a(n)=A001906(n)*A000244(n-1)=A001906(n)*A000244(n)/3. - Robert G. Wilson 
               v Sep 21 2006
%F A057085 a(2k)=A004187(k)*9^k/3, a(2k-1)= A033890(k)*9^k.
%F A057085 G.f.: x/(1-9*x+9*x^2).
%F A057085 a(n)=(1/3)*sum(k=0, n, binomial(n, k)*F(4*k)) where F(k) denotes the 
               k-th Fibonacci number. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jun 21 2003
%F A057085 a(n)=-(1/15)*[9/2-(3/2)*sqrt(5)]^n*sqrt(5)+(1/15)*sqrt(5)*[9/2+(3/2)*sqrt(5)]^n 
               - Paolo P. Lava (ppl(AT)spl.at), Jun 16 2008
%F A057085 a(n+1)=Sum_{k, 0<=k<=n} A109466(n,k)*9^k. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 28 2008]
%t A057085 f[n_] := Fibonacci[2n]*3^(n - 1); Table[f@n, {n, 0, 20}] (* or *)
%t A057085 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 A057085 CoefficientList[Series[x/(1 - 9x + 9x^2), {x, 0, 20}], x] (* Robert G. 
               Wilson v Sep 21 2006 *)
%o A057085 (PARI) a(n)=(1/3)*sum(k=0,n,binomial(n,k)*fibonacci(4*k)) (Cloitre)
%o A057085 (Other) sage: [lucas_number1(n,9,9) for n in xrange(0, 21)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
%Y A057085 Cf. A030191.
%Y A057085 Sequence in context: A162755 A045993 A084327 this_sequence A076765 A006634 
               A129328
%Y A057085 Adjacent sequences: A057082 A057083 A057084 this_sequence A057086 A057087 
               A057088
%K A057085 nonn,easy
%O A057085 0,3
%A A057085 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000
%E A057085 Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 16 2005.

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