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A057085
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a(n) = 9*a(n-1) - 9*a(n-2) for n>1, with a(0)=0, a(1)=1.
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12
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0, 1, 9, 72, 567, 4455, 34992, 274833, 2158569, 16953624, 133155495, 1045816839, 8213952096, 64513217313, 506693386953, 3979621526760, 31256353258263, 245490585583527, 1928108090927376, 15143557548094641, 118939045114505385, 934159388097696696
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OFFSET
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0,3
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COMMENTS
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Scaled Chebyshev U-polynomials evaluated at 3/2.
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LINKS
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FORMULA
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a(n) = S(n, 3)*3^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
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
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MATHEMATICA
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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 *)
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PROG
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(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
(Magma) [3^(n-1)*Fibonacci(2*n): n in [0..30]]; // G. C. Greubel, May 02 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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