A030195 a(n) = 3*a(n-1) + 3*a(n-2), a(0)=0, a(1)=1.

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

Offset: 0

Author

Wolfdieter Lang

Keywords

• nonn
• easy

Comments

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

Number of zeros in the substitution system {0 -> 1111100, 1 -> 10} at step n from initial string "1" (1 -> 10 -> 101111100 -> ...). - Ilya Gutkovskiy, Apr 10 2017

a(n+1) is the number of compositions of n having parts 1 and 2, both of three kinds. - Gregory L. Simay, Sep 21 2017

More generally, define a(n) = k*a(n-1) + k*a(n-2), a(0) = 0 and a(1) = 1. Then g.f. a(n) = 1/(1 - k*x - k*x^2) and a(n+1) is the number of compositions of n having parts 1 and 2, both of k kinds. - Gregory L. Simay, Sep 22 2017

Examples

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

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).

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.

Programs

Haskell

a030195 n = a030195_list !! n
a030195_list =
   0 : 1 : map (* 3) (zipWith (+) a030195_list (tail a030195_list))
-- Reinhard Zumkeller, Oct 14 2011

Magma

I:=[0,1]; [n le 2 select I[n] else 3*Self(n-1) + 3*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018

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 *)

PARI

{a(n) = n--; polchebyshev(n, 2, I*sqrt(3)/2) * (-I*sqrt(3))^n};

Sage

[lucas_number1(n,3,-3) for n in range(0, 25)] # Zerinvary Lajos, Apr 22 2009

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_{k=0..floor(n/2)} 3^(n-k)*binomial(n-k, k). - 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

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