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A030195
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a(n) = 3*a(n-1)+3*a(n-2), a(0)=0, a(1)=1.
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39
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0, 1, 3, 12, 45, 171, 648, 2457, 9315, 35316, 133893, 507627, 1924560, 7296561, 27663363, 104879772, 397629405, 1507527531, 5715470808, 21668995017, 82153397475, 311467177476, 1180861724853, 4476986706987, 16973545295520
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Scaled Chebyshev U-polynomials evaluated at I*sqrt(3)/2.
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REFERENCES
| 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=q=3.
Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", Wiley, 2001, p. 471.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (39), (41) and (45), rhs, m=3.
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,3).
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FORMULA
| a(n+1)=(-I*sqrt(3))^n*U(n, I*sqrt(3)/2), g.f.: x/(1-3*x-3*x^2).
a(n+1) = sum(3^(n-k)*binomial(n-k, k), k=0..floor(n/2)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 14 2001
a(n) = (p^n - q^n)/sqrt(21); p = (3 + sqrt 21)/2, q = (3 - sqrt 21)/2. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 02 2003
For n > 0, a(n) = Sum_{k=0..n-1} (2^k)*A063967(n-1,k) - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Jul 23 2006
a(n+1)=Sum_{k, 0<=k<=n}2^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2006
G.f.: x/(1-3x-3x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 19 2008]
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MATHEMATICA
| CoefficientList[Series[1/(1-3x-3x^2), {x, 0, 25}], x] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 22 2007
f[n_, 0] = 0; f[n_, 1] = 1; f[n_, i_] := 3 f[n, i - 1] + 3 f[n, i - 2]; Table[f[3, i], {i, 0, 24}] [From Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 20 2010]
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PROG
| (sage) [lucas_number1(n, 3, -3) for n in xrange(0, 25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
(PARI) a(n)=n--; polchebyshev(n, 2, I*sqrt(3)/2)*(-I*sqrt(3))^n
(Haskell)
a030195 n = a030195_list !! n
a030195_list =
0 : 1 : map (* 3) (zipWith (+) a030195_list (tail a030195_list))
-- Reinhard Zumkeller, Oct 14 2011
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CROSSREFS
| Equals round(A085480(n)/sqrt(21)).
Cf. A000045, A002605, A172010, A057088, A057089, A057090, A057091, A057092, A057093. [From Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 20 2010]
Cf. A026150, A028859, A028860, A080040, A083337, A106435, A108898, A125145.
Sequence in context: A062561 A128593 A085481 * A114515 A192467 A151162
Adjacent sequences: A030192 A030193 A030194 * A030196 A030197 A030198
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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EXTENSIONS
| Edited by Ralf Stephan, Aug 02 2004
I simplified the definition. As a result the offsets in some of the formulae may need to shifted by 1. - N. J. A. Sloane (njas(AT)research.att.com), Apr 01, 2006.
Formulas shifted to match offset. - Charles R Greathouse IV, Jan 31 2011
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