A122552 a(0)=a(1)=a(2)=1, a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n > 2.

1, 1, 1, 4, 7, 13, 28, 55, 109, 220, 439, 877, 1756, 3511, 7021, 14044, 28087, 56173, 112348, 224695, 449389, 898780, 1797559, 3595117, 7190236, 14380471, 28760941, 57521884, 115043767, 230087533, 460175068, 920350135, 1840700269, 3681400540

Offset

0,4

Comments

Equals INVERT transform of (1, 0, 3, 0, 3, 0, 3, ...). - Gary W. Adamson, Apr 27 2009

No term is divisible by 3. - Vladimir Joseph Stephan Orlovsky, Mar 24 2011

For n > 3, a(n) is the number of quaternary sequences of length n-1 starting with q(0) = 0, in which all triples (q(i), q(i+1), q(i+2)) contain digits 0 and 3; cf. A294627. - Wojciech Florek, Jul 30 2018

For n > 0, a(n) is the number of ways to tile a strip of length n with squares, dominoes, and two colors of trominoes, with the restriction that the first tile cannot be a domino. - Greg Dresden and Bora Bursalı, Aug 31 2023

Links

Wojciech Florek, Table of n, a(n) for n = 0..2000

Wojciech Florek, A class of generalized Tribonacci sequences applied to counting problems, Appl. Math. Comput., 338 (2018), 809-821.

Index entries for linear recurrences with constant coefficients, signature (1,1,2).

Formula

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

G.f. : (1-x^2)/(1-x-x^2-2*x^3).

a(n) = (3/7)*2^n - (1/7)*i*sqrt(3)*((-1/2) + (1/2)*i*sqrt(3))^n + (2/7)*((-1/2) - (1/2)*i*sqrt(3))^n + (2/7)*((-1/2) + (1/2)*i*sqrt(3))^n + (1/7)*i*sqrt(3)*((-1/2) - (1/2)*i*sqrt(3))^n, with n >= 0 and i=sqrt(-1). - Paolo P. Lava, Nov 19 2008

a(n) = ((-1)^n*A130815(n+2) + 3*2^n)/7. -R. J. Mathar, Nov 30 2008

From Paul Curtz, Oct 02 2009: (Start)

a(n) = A140295(n+2)/4.

a(n+1) - 2a(n) = period 3: repeat -1,-1,2 = -A061347.

a(n) - a(n-1) = 0,0,3,3,6,15,27,54,111,... = 3*A077947.

a(n) - a(n-2) = 0,3,6,9,21,42,81,....

a(n) - a(n-3) = 3,6,12,24,... = A007283 = 3*A000079.

a(3n) + a(3n+1) + a(3n+2) = 3,24,192,... = A103333(n+1) = A140295(3n) + A140295(3n+1) + A140295(3n+2).

See A078010, A139217, A139218. (End)

Example

It is shown in A294627 that there are 42 quaternary sequences (i.e., build from four digits 0, 1, 2, 3) and having both 0 and 3 in every (consecutive) triple. Only a(5=4+1) = 13 of them start with 0: 003x, 030x, 03y0, 0y30, 0330, where x = 0, 1, 2, 3 and y = 1, 2.

Maple

seq(coeff(series((1-x^2)/(1-x-x^2-2*x^3), x, n+1), x, n), n=0..40); # Muniru A Asiru, Aug 02 2018

Mathematica

LinearRecurrence[{1, 1, 2}, {1, 1, 1}, 40]
CoefficientList[ Series[(x^2 - 1)/(2x^3 + x^2 + x - 1), {x, 0, 35}], x] (* Robert G. Wilson v, Jul 30 2018 *)

Prog

(Sage) from sage.combinat.sloane_functions import recur_gen3; it = recur_gen3(1, 1, 1, 1, 1, 2); [next(it) for i in range(30)] # Zerinvary Lajos, Jun 25 2008
(PARI) Vec((1-x^2)/(1-x-x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Jan 17 2012
(GAP) a:=[1, 1, 1];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+2*a[n-3]; od; a; # Muniru A Asiru, Jul 30 2018

Crossrefs

Cf. A294627.

Sequence in context: A304004 A293342 A298346 * A197546 A074862 A101064

Adjacent sequences: A122549 A122550 A122551 * A122553 A122554 A122555

Keyword

nonn,easy

Author

Philippe Deléham, Sep 20 2006

Extensions

Corrected by T. D. Noe, Nov 01 2006, Nov 07 2006

Typo in definition corrected by Paul Curtz, Oct 02 2009

Status

approved