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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.

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.

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).

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<=k<=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 formulae 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 November 20 21:20 EST 2014. Contains 249754 sequences.