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%I #104 Feb 25 2023 14:17:11
%S 1,3,6,9,9,0,-27,-81,-162,-243,-243,0,729,2187,4374,6561,6561,0,
%T -19683,-59049,-118098,-177147,-177147,0,531441,1594323,3188646,
%U 4782969,4782969,0,-14348907,-43046721,-86093442,-129140163,-129140163,0
%N Scaled Chebyshev U-polynomials evaluated at sqrt(3)/2; expansion of 1/(1 - 3*x + 3*x^2).
%C With different sign pattern, see A000748.
%C 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
%H Robert Israel, <a href="/A057083/b057083.txt">Table of n, a(n) for n = 0..4170</a>
%H T. Alden Gassert, <a href="http://arxiv.org/abs/1409.7829">Discriminants of simplest 3^n-tic extensions</a>, arXiv preprint arXiv:1409.7829 [math.NT], 2014.
%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=3, q=-3.
%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%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=3.
%H Vladimir Shevelev, <a href="https://arxiv.org/abs/1706.01454">Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n</a>, arXiv:1706.01454 [math.CO], 2017.
%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 (3,-3).
%F a(n) = S(n, sqrt(3))*(sqrt(3))^n with S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.
%F a(2*n) = A057078(n)*3^n; a(2*n+1)= A010892(n)*3^(n+1).
%F G.f.: 1/(1-3*x+3*x^2).
%F Binomial transform of A057079. a(n) = Sum_{k=0..n} 2*binomial(n, k)*cos((k-1)Pi/3). - _Paul Barry_, Aug 19 2003
%F For n > 5, a(n) = -27*a(n-6) - _Gerald McGarvey_, Apr 21 2005
%F a(n) = Sum_{k=0..n} A109466(n,k)*3^k. - _Philippe Deléham_, Nov 12 2008
%F 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
%F 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
%F 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
%F 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
%F 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
%F From _Vladimir Shevelev_, Jul 30 2017: (Start)
%F a(n) = 2*3^(n/2)*cos(Pi*(n-2)/6);
%F a(n) = K_2(n+2) - K_1(n+2);
%F 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)
%p seq(3^(n/2)*orthopoly[U](n,sqrt(3)/2),n=0..100); # _Robert Israel_, Nov 21 2016
%t Join[{a=1,b=3},Table[c=3*b-3*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 17 2011 *)
%t CoefficientList[Series[1/(1 - 3 x + 3 x^2), {x, 0, 35}], x] (* _Michael De Vlieger_, Jul 30 2017 *)
%o (Sage) [lucas_number1(n,3,3) for n in range(1, 37)] # _Zerinvary Lajos_, Apr 23 2009
%o (PARI) a(n)=([0,1; -3,3]^n*[1;3])[1,1] \\ _Charles R Greathouse IV_, Apr 08 2016
%o (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
%Y Cf. A000748, A010892, A049310, A057078, A057681, A057682, A129339.
%K easy,sign
%O 0,2
%A _Wolfdieter Lang_, Aug 11 2000
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