A057085 a(n) = 9*a(n-1) - 9*a(n-2) for n>1, with a(0)=0, a(1)=1.

0, 1, 9, 72, 567, 4455, 34992, 274833, 2158569, 16953624, 133155495, 1045816839, 8213952096, 64513217313, 506693386953, 3979621526760, 31256353258263, 245490585583527, 1928108090927376, 15143557548094641, 118939045114505385, 934159388097696696

Offset: 0

Author

Wolfdieter Lang, Aug 11 2000

Keywords

• nonn
• easy

Comments

Scaled Chebyshev U-polynomials evaluated at 3/2.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• 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.
• Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(38) and (45),lhs, m=9.
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (9,-9).

Crossrefs

Cf. A000045, A000244, A001906, A004187, A030191, A033890, A049310, A109466.

Programs

Magma

[3^(n-1)*Fibonacci(2*n): n in [0..30]]; // G. C. Greubel, May 02 2022

Mathematica

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)) \\ Benoit Cloitre

PARI

concat(0, Vec(x/(1-9*x+9*x^2) + O(x^30))) \\ Colin Barker, Jun 14 2015

Sage

[lucas_number1(n,9,9) for n in range(0, 21)] # Zerinvary Lajos, Apr 23 2009

Formula

a(n) = A001906(n)*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)/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)*Fibonacci(4*k). - Benoit Cloitre, Jun 21 2003

a(n+1) = Sum_{k=0..n} A109466(n,k)*9^k. - Philippe Deléham, Oct 28 2008

Extensions

Edited by N. J. A. Sloane, Sep 16 2005