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A057083 Scaled Chebyshev U-polynomials evaluated at sqrt(3)/2; expansion of 1/(1-3*x+3*x^2). 33


%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 a(n) = 6a(n-1)-15a(n-2)+20a(n-3)-15a(n-4)+6a(n-5). - _Paul Curtz_, Nov 21 2007

%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 T. Alden Gassert, <a href="http://arxiv.org/abs/1409.7829">Discriminants of simplest 3^n-tic extensions</a>, arXiv preprint arXiv:1409.7829, 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 Victorovich Kruchinin , <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2011.

%H W. 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 <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*C(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

%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 *)

%o (Sage) [lucas_number1(n,3,3) for n in xrange(1, 37)] # _Zerinvary Lajos_, Apr 23 2009

%Y Cf. A049310, A057078, A010892, A000748.

%Y Cf. A129339, A057681, A057682.

%K easy,sign

%O 0,2

%A _Wolfdieter Lang_, Aug 11 2000

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Last modified August 29 01:49 EDT 2015. Contains 261184 sequences.