login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A057083 Scaled Chebyshev U-polynomials evaluated at sqrt(3)/2; expansion of 1/(1 - 3*x + 3*x^2). 34

%I

%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 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 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*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

%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 *)

%o (Sage) [lucas_number1(n,3,3) for n in xrange(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

%Y Cf. A049310, A057078, A010892, A000748, A129339, A057681, A057682.

%K easy,sign,changed

%O 0,2

%A _Wolfdieter Lang_, Aug 11 2000

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 3 16:24 EST 2016. Contains 278745 sequences.