0,3

Scaled Chebyshev U-polynomials evaluated at 3/2.

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=9, q=-9.

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=9.

Index entries for sequences related to Chebyshev polynomials.

Also Fibonacci(2n)*3^(n-1).

a(n) = S(n, 3)*3^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.

a(n)=A001906(n)*A000244(n-1)=A001906(n)*A000244(n)/3. - Robert G. Wilson v Sep 21 2006

a(2k)=A004187(k)*9^k/3, a(2k-1)= A033890(k)*9^k.

G.f.: x/(1-9*x+9*x^2).

a(n)=(1/3)*sum(k=0, n, binomial(n, k)*F(4*k)) where F(k) denotes the k-th Fibonacci number. - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 21 2003

a(n)=-(1/15)*[9/2-(3/2)*sqrt(5)]^n*sqrt(5)+(1/15)*sqrt(5)*[9/2+(3/2)*sqrt(5)]^n - Paolo P. Lava (ppl(AT)spl.at), Jun 16 2008

a(n+1)=Sum_{k, 0<=k<=n} A109466(n,k)*9^k. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 28 2008]

f[n_] := Fibonacci[2n]*3^(n - 1); Table[f@n, {n, 0, 20}] (* or *)

a[0] = 0; a[1] = 1; a[n_] := a[n] = 9(a[n - 1] - a[n - 2]); Table[a[n], {n, 0, 20}] (* or *)

CoefficientList[Series[x/(1 - 9x + 9x^2), {x, 0, 20}], x] (* Robert G. Wilson v Sep 21 2006 *)

(PARI) a(n)=(1/3)*sum(k=0, n, binomial(n, k)*fibonacci(4*k)) (Cloitre)

(Other) sage: [lucas_number1(n, 9, 9) for n in xrange(0, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]

Cf. A030191.

Sequence in context: A110396 A045993 A084327 this_sequence A076765 A006634 A129328

Adjacent sequences: A057082 A057083 A057084 this_sequence A057086 A057087 A057088

nonn,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000

Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 16 2005.