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%I #43 Sep 08 2022 08:45:27
%S 1,3,9,24,66,180,492,1344,3672,10032,27408,74880,204576,558912,
%T 1526976,4171776,11397504,31138560,85072128,232421376,634987008,
%U 1734816768,4739607552,12948848640,35376912384,96651522048,264056868864,721416781824,1970947301376
%N Expansion of g.f.: (1 + x + x^2)/(1 - 2*x - 2*x^2).
%C a(n-1) is the number of compositions of n into floor((3*j-1)/2) kinds of j's for all j >= 1. The sequence of such compositions is 1,1,3,9,24,... (i.e., this sequence prepended by 1) and has g.f. 1/(1-Sum_{j>=1} floor((3*j-1)/2)*x^j). - _Joerg Arndt_, Jul 06 2011
%C a(n) is the number of length n words on 3 letters (ternary words) such that the length of any run of identical letters is <= 2. Cf. A181137 for a generalization. - _Geoffrey Critzer_, Sep 16 2013
%D A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14.
%H G. C. Greubel, <a href="/A121907/b121907.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Burstein and T. Mansour, <a href="https://arxiv.org/abs/math/0112281">Words restricted by 3-letter generalized multipermutation patterns</a>, arXiv:math/0112281 [math.CO], 2001.
%H Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="https://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,2).
%F a(0)=1, a(1)=3, a(2)=9; a(n) = 2*a(n-1) + 2*a(n-2) for n>=3. - _Philippe Deléham_, Sep 19 2009
%F G.f.: (1/x)*(-1 + 1/(1-Sum_{j>=1} floor((3*j-1)/2)*x^j)). - _Joerg Arndt_, Jul 06 2011
%F E.g.f.: (1/2)*exp(x)*(3*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - 1/2. - _Stefano Spezia_, Oct 08 2019
%p seq(coeff(series((1+x+x^2)/(1-2*x-2*x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Oct 07 2019
%t CoefficientList[Series[(1+x+x^2)/(1-2x-2x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,2},{1,3,9},30] (* _Harvey P. Dale_, Dec 03 2011 *)
%o (PARI) my(x='x+O('x^30)); Vec((1+x+x^2)/(1-2*x-2*x^2)) \\ _G. C. Greubel_, Oct 07 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x+x^2)/(1-2*x-2*x^2) )); // _G. C. Greubel_, Oct 07 2019
%o (Sage)
%o def A121907_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((1+x+x^2)/(1-2*x-2*x^2)).list()
%o A121907_list(30) # _G. C. Greubel_, Oct 07 2019
%o (GAP) a:=[3,9];; for n in [3..30] do a[n]:=2*(a[n-1]+a[n-2]); od; Concatenation([1], a); # _G. C. Greubel_, Oct 07 2019
%Y Column 3 in A265584.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Nov 20 2006
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