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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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43
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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, seeA000748.
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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LINKS
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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.
G.f.: 1/(1-3*x+3*x^2).
Binomial transform ofA057079.a(n) = Sum_{k=0..n} 2*binomial(n, k)*cos((k-1)Pi/3). -Paul Barry,Aug 19 2003
a(n) = Sum_{k=1..n} binomial(k,n-k) * 3^k *(-1)^(n-k) for n>0; a(0)=1. -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
G.f.: G(0)/(2-3*x), where G(k)= 1 + 1/(1 - x*(k+3)/(x*(k+4) + 2/G(k+1))); (continued fraction). -Sergei N. Gladkovskii,Jun 16 2013
a(n) = Sum_{k = 0..floor(n/3)} (-1)^k*binomial(n+2,3*k+2). Sykora's conjecture in the Comments section follows easily from this. -Peter Bala,Nov 21 2016
a(n) = 2*3^(n/2)*cos(Pi*(n-2)/6);
a(n) = K_2(n+2) - K_1(n+2);
For m,n>=1, a(n+m) = a(n-1)*K_1(m+1) + K_2(n+1)*K_2(m+1) + K_1(n+1)*a(m-1) where K_1 =A057681,K_2 =A057682.(End)
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MAPLE
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seq(3^(n/2)*orthopoly[U](n, sqrt(3)/2), n=0..100); #Robert Israel,Nov 21 2016
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MATHEMATICA
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CoefficientList[Series[1/(1 - 3 x + 3 x^2), {x, 0, 35}], x] (*Michael De Vlieger,Jul 30 2017 *)
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PROG
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(Sage) [lucas_number1(n, 3, 3) for n in range(1, 37)] #Zerinvary Lajos,Apr 23 2009
(Magma) I:=[1, 3]; [n le 2 select I[n] else 3*Self(n-1) - 3*Self(n-2): n in [1..30]]; //G. C. Greubel,Oct 23 2018
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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