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URL: https://oeis.org/A057085

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A057085
a(n) = 9*a(n-1) - 9*a(n-2) for n>1, with a(0)=0, a(1)=1.
13
0, 1, 9, 72, 567, 4455, 34992, 274833, 2158569, 16953624, 133155495, 1045816839, 8213952096, 64513217313, 506693386953, 3979621526760, 31256353258263, 245490585583527, 1928108090927376, 15143557548094641, 118939045114505385, 934159388097696696
OFFSET
0,3
COMMENTS
Scaled Chebyshev U-polynomials evaluated at 3/2.
LINKS
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.
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
E.g.f.: 2*exp(9*x/2)*sinh(3*sqrt(5)*x/2)/(3*sqrt(5)). - Stefano Spezia, Sep 01 2025
27*a(n) + 3 = Sum_{k=0..n+1} 3^k*Fibonacci(2*k). - Greg Dresden and Xiaoya Gao, Nov 24 2025
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 *)
PROG
(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
(SageMath) [lucas_number1(n, 9, 9) for n in range(0, 21)] # Zerinvary Lajos, Apr 23 2009
(Magma) [3^(n-1)*Fibonacci(2*n): n in [0..30]]; // G. C. Greubel, May 02 2022
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Aug 11 2000
EXTENSIONS
Edited by N. J. A. Sloane, Sep 16 2005
STATUS
approved