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A057083
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Scaled Chebyshev U-polynomials evaluated at sqrt(3)/2; expansion of 1/(1-3*x+3*x^2).
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30
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1, 3, 6, 9, 9, 0, -27, -81, -162, -243, -243, 0, 729, 2187, 4374, 6561, 6561, 0, -19683, -59049, -118098, -177147, -177147, 0, 531441, 1594323, 3188646, 4782969, 4782969, 0, -14348907, -43046721, -86093442, -129140163, -129140163, 0
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OFFSET
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0,2
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COMMENTS
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With different sign pattern, see A000748.
a(n)=6a(n-1)-15a(n-2)+20a(n-3)-15a(n-4)+6a(n-5). - Paul Curtz, Nov 21 2007
Conjecture: Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1-M)^n = A057681(n)-A057682(n)*M+z(n)*M^2, where z(0)=z(1)=0 and, apparently, z(n+2)=a(n). - Stanislav Sykora, Jun 10 2012
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REFERENCES
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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.
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.
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LINKS
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Table of n, a(n) for n=0..35.
Index entries for sequences related to Chebyshev polynomials.
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565
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FORMULA
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a(n)=S(n, sqrt(3))*(sqrt(3))^n with S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.
a(2*n)= A057078(n)*3^n; a(2*n+1)= A010892(n)*3^(n+1).
G.f.: 1/(1-3*x+3*x^2).
Binomial transform of A057079. a(n)=sum{k=0..n, 2*C(n, k)*cos((k-1)pi/3) }. - Paul Barry, Aug 19 2003
For n > 5, a(n) = -27*a(n-6) - Gerald McGarvey, Apr 21 2005
a(n)=Sum_{k, 0<=k<=n}A109466(n,k)*3^k. [From Philippe DELEHAM, Nov 12 2008]
a(n) = sum(k=1..n, binomial(k,n-k) * 3^(k)*(-1)^(n-k)) for n>0; a(0)=1. [From Vladimir Kruchinin, Feb 07 2011]
By the conjecture: Start with x(0)=1,y(0)=0,z(0)=0 and set x(n+1)=x(n)-z(n), y(n+1)=y(n)-x(n),z(n+1)=z(n)-y(n). Then a(n)=z(n+2). This recurrence indeed ends up in a repetitive cycle of length 6 and multiplicative factor -27, confirming G.McGarvey's observation. - Stanislav Sykora, Jun 10 2012
G.f.: Q(0) where Q(k) = 1 + k*(3*x+1) + 9*x - 3*x*(k+1)*(k+4)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Mar 15 2013
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MATHEMATICA
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Join[{a=1, b=3}, Table[c=3*b-3*a; a=b; b=c, {n, 100}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 17 2011*)
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PROG
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(Sage) [lucas_number1(n, 3, 3) for n in xrange(1, 37)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
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CROSSREFS
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A049310, A057078, A010892, A000748.
Cf. A129339, A057681, A057682.
Sequence in context: A021077 A114041 A212712 * A000748 A198373 A160178
Adjacent sequences: A057080 A057081 A057082 * A057084 A057085 A057086
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KEYWORD
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easy,sign
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AUTHOR
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Wolfdieter Lang, Aug 11 2000
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STATUS
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approved
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