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A057085
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a(0)=0, a(1)=1; for n>1, a(n) = 9a(n-1) - 9a(n-2).
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8
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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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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.
W. 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).
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FORMULA
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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, 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, Jun 16 2008
a(n+1) = Sum_{k=0..n} A109466(n,k)*9^k. - Philippe Deléham, Oct 28 2008
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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
(Sage) [lucas_number1(n, 9, 9) for n in xrange(0, 21)] # Zerinvary Lajos, Apr 23 2009
(PARI) concat(0, Vec(x/(1-9*x+9*x^2) + O(x^30))) \\ Colin Barker, Jun 14 2015
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CROSSREFS
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Cf. A000045, A030191.
Sequence in context: A162755 A045993 A084327 * A076765 A006634 A129328
Adjacent sequences: A057082 A057083 A057084 * A057086 A057087 A057088
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Aug 11 2000
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EXTENSIONS
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Edited by N. J. A. Sloane, Sep 16 2005
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STATUS
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approved
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