This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A122552 #60 Dec 18 2023 12:16:31
%S A122552 1,1,1,4,7,13,28,55,109,220,439,877,1756,3511,7021,14044,28087,56173,
%T A122552 112348,224695,449389,898780,1797559,3595117,7190236,14380471,
%U A122552 28760941,57521884,115043767,230087533,460175068,920350135,1840700269,3681400540
%N A122552 a(0)=a(1)=a(2)=1, a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n > 2.
%C A122552 Equals INVERT transform of (1, 0, 3, 0, 3, 0, 3, ...). - _Gary W. Adamson_, Apr 27 2009
%C A122552 No term is divisible by 3. - _Vladimir Joseph Stephan Orlovsky_, Mar 24 2011
%C A122552 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
%C A122552 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
%H A122552 Wojciech Florek, <a href="/A122552/b122552.txt">Table of n, a(n) for n = 0..2000</a>
%H A122552 Wojciech Florek, <a href="http://doi.org/10.1016/j.amc.2018.06.014">A class of generalized Tribonacci sequences applied to counting problems</a>, Appl. Math. Comput., 338 (2018), 809-821.
%H A122552 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,2).
%F A122552 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.
%F A122552 G.f.: (1-x^2)/(1-x-x^2-2*x^3).
%F A122552 a(n) = ((-1)^n*A130815(n+2) + 3*2^n)/7. - _R. J. Mathar_, Nov 30 2008
%F A122552 From _Paul Curtz_, Oct 02 2009: (Start)
%F A122552 a(n) = A140295(n+2)/4.
%F A122552 a(n+1) - 2a(n) = period 3: repeat -1,-1,2 = -A061347.
%F A122552 a(n) - a(n-1) = 0,0,3,3,6,15,27,54,111,... = 3*A077947.
%F A122552 a(n) - a(n-2) = 0,3,6,9,21,42,81,....
%F A122552 a(n) - a(n-3) = 3,6,12,24,... = A007283 = 3*A000079.
%F A122552 a(3n) + a(3n+1) + a(3n+2) = 3,24,192,... = A103333(n+1) = A140295(3n) + A140295(3n+1) + A140295(3n+2).
%F A122552 See A078010, A139217, A139218. (End)
%e A122552 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.
%p A122552 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
%t A122552 LinearRecurrence[{1, 1, 2}, {1, 1, 1}, 40]
%t A122552 CoefficientList[ Series[(x^2 - 1)/(2x^3 + x^2 + x - 1), {x, 0, 35}], x] (* _Robert G. Wilson v_, Jul 30 2018 *)
%o A122552 (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
%o A122552 (PARI) Vec((1-x^2)/(1-x-x^2-2*x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Jan 17 2012
%o A122552 (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
%Y A122552 Cf. A294627.
%K A122552 nonn,easy
%O A122552 0,4
%A A122552 _Philippe Deléham_, Sep 20 2006
%E A122552 Corrected by _T. D. Noe_, Nov 01 2006, Nov 07 2006
%E A122552 Typo in definition corrected by _Paul Curtz_, Oct 02 2009