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A030195 a(n) = 3*a(n-1) + 3*a(n-2), a(0)=0, a(1)=1. 44
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; text; internal format)
OFFSET

0,3

COMMENTS

Scaled Chebyshev U-polynomials evaluated at I*sqrt(3)/2.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

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.

Tanya Khovanova, Recursive Sequences

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (39), (41) and (45), rhs, m=3.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (3,3).

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, Nov 14 2001

a(n) = (p^n - q^n)/sqrt(21); p = (3 + sqrt 21)/2, q = (3 - sqrt 21)/2. - Gary W. Adamson, Jul 02 2003

For n > 0, a(n) = Sum_{k=0..n-1} (2^k)*A063967(n-1,k) - Gerald McGarvey, Jul 23 2006

a(n+1) = Sum_{k=0..n} 2^k*A063967(n,k). - Philippe Deléham, Nov 03 2006

G.f.: x/(1-3x-3x^2). - Philippe Deléham, Nov 19 2008

EXAMPLE

G.f. = x + 3*x^2 + 12*x^3 + 45*x^4 + 171*x^5 + 648*x^6 + 2457*x^7 + ...

MATHEMATICA

CoefficientList[Series[1/(1-3x-3x^2), {x, 0, 25}], x] (* Zerinvary Lajos, Mar 22 2007 *)

LinearRecurrence[{3, 3}, {0, 1}, 24] (* Or *)

RecurrenceTable[{a[n] == 3 a[n - 1] + 3 a[n - 2], a[0] == 0, a[1] == 1}, a, {n, 0, 23}] (* Robert G. Wilson v, Aug 18 2012 *)

PROG

(Sage) [lucas_number1(n, 3, -3) for n in xrange(0, 25)] # Zerinvary Lajos, 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

CROSSREFS

Equals round(A085480(n)/sqrt(21)).

Cf. A175290 (Pisano periods), A000045, A002605, A172010, A057088, A057089, A057090, A057091, A057092, A057093.

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

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang

EXTENSIONS

Edited by Ralf Stephan, Aug 02 2004

I simplified the definition. As a result the offsets in some of the formulas may need to shifted by 1. - N. J. A. Sloane, Apr 01 2006

Formulas shifted to match offset. - Charles R Greathouse IV, Jan 31 2011

STATUS

approved

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Last modified December 7 03:42 EST 2016. Contains 278840 sequences.