|
| |
|
|
A057083
|
|
Scaled Chebyshev U-polynomials evaluated at sqrt(3)/2; expansion of 1/(1-3*x+3*x^2).
|
|
27
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| 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 (bpcrtz(AT)free.fr), Nov 21 2007
|
|
|
REFERENCES
| 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.
|
|
|
LINKS
| Index entries for sequences related to Chebyshev polynomials.
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565
|
|
|
FORMULA
| 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 (pbarry(AT)wit.ie), Aug 19 2003
For n > 5, a(n) = -27*a(n-6) - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Apr 21 2005
a(n)=Sum_{k, 0<=k<=n}A109466(n,k)*3^k. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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 (kru(AT)ie.tusur.ru), Feb 07 2011]
|
|
|
MATHEMATICA
| 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*)
|
|
|
PROG
| (Other) sage: [lucas_number1(n, 3, 3) for n in xrange(1, 37)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
|
|
|
CROSSREFS
| A049310, A057078, A010892, A000748.
Cf. A129339.
Sequence in context: A137991 A021077 A114041 * A000748 A198373 A160178
Adjacent sequences: A057080 A057081 A057082 * A057084 A057085 A057086
|
|
|
KEYWORD
| easy,sign
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 11 2000
|
| |
|
|
