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 A122552 a(0)=a(1)=a(2)=1, a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n > 2. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 LINKS Wojciech Florek, Table of n, a(n) for n = 0..2000 W. 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

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Last modified April 14 12:11 EDT 2021. Contains 342949 sequences. (Running on oeis4.)