Search: id:A057083
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%I A057083
%S A057083 1,3,6,9,9,0,27,81,162,243,243,0,729,2187,4374,6561,6561,0,19683,59049,
%T A057083 118098,177147,177147,0,531441,1594323,3188646,4782969,4782969,0,
%U A057083 14348907,43046721,86093442,129140163,129140163,0
%V A057083 1,3,6,9,9,0,-27,-81,-162,-243,-243,0,729,2187,4374,6561,6561,0,
%W A057083 -19683,-59049,-118098,-177147,-177147,0,531441,1594323,3188646,
%X A057083 4782969,4782969,0,-14348907,-43046721,-86093442,-129140163,-129140163,
               0
%N A057083 Scaled Chebyshev U-polynomials evaluated at sqrt(3)/2; expansion of 1/
               (1-3*x+3*x^2).
%C A057083 With different sign pattern, see A000748.
%C A057083 a(n)=6a(n-1)-15a(n-2)+20a(n-3)-15a(n-4)+6a(n-5). - Paul Curtz (bpcrtz(AT)free.fr), 
               Nov 21 2007
%D A057083 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=3, q=-3.
%D A057083 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=3.
%H A057083 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057083 a(n)=S(n, sqrt(3))*(sqrt(3))^n with S(n, x) := U(n, x/2), Chebyshev polynomials 
               of 2nd kind, A049310.
%F A057083 a(2*n)= A057078(n)*3^n; a(2*n+1)= A010892(n)*3^(n+1).
%F A057083 G.f.: 1/(1-3*x+3*x^2).
%F A057083 Binomial transform of A057079. a(n)=sum{k=0..n, 2*C(n, k)*cos((k-1)pi/
               3) }. - Paul Barry (pbarry(AT)wit.ie), Aug 19 2003
%F A057083 For n > 5, a(n) = -27*a(n-6) - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), 
               Apr 21 2005
%F A057083 a(n)=Sum_{k, 0<=k<=n}A109466(n,k)*3^k. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 12 2008]
%o A057083 (Other) sage: [lucas_number1(n,3,3) for n in xrange(1, 37)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
%Y A057083 A049310, A057078, A010892, A000748.
%Y A057083 Cf. A129339.
%Y A057083 Sequence in context: A137991 A021077 A114041 this_sequence A000748 A160178 
               A011383
%Y A057083 Adjacent sequences: A057080 A057081 A057082 this_sequence A057084 A057085 
               A057086
%K A057083 easy,sign
%O A057083 0,2
%A A057083 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 11 
               2000

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