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%I #31 Jul 21 2018 08:15:46
%S 1,2,9,36,144,576,2304,9216,36864,147456,589824,2359296,9437184,
%T 37748736,150994944,603979776,2415919104,9663676416,38654705664,
%U 154618822656,618475290624,2473901162496,9895604649984,39582418599936,158329674399744,633318697598976,2533274790395904,10133099161583616,40532396646334464,162129586585337856,648518346341351424
%N Number of compositions of n into 3*j-1 kinds of j's for all j >= 1.
%C First differences of A002001.
%C For n >= 2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4} we have f(x_1) <> y_1 and f(x_2) <> y_2. - _Milan Janjic_, Apr 19 2007
%C Convolved with [1, 2, 3, ...] = powers of 4: [1, 4, 16, 64, ...]. - _Gary W. Adamson_, Jun 04 2009
%C a(n) is the number of generalized compositions of n when there are 3 *i-1 different types of i, (i=1,2,...). - _Milan Janjic_, Aug 26 2010
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H Daniel Birmajer, Juan B. Gil, Michael D. Weiner, <a href="https://arxiv.org/abs/1707.07798">(an + b)-color compositions</a>, arXiv:1707.07798 [math.CO], 2017.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (4).
%F a(n) = 9*4^(n-2), a(0)=1, a(1)=2.
%F a(0)=1, a(1)=2, a(3)=9, a(n+1)=4*a(n) for n >= 3.
%F G.f.: (1-x)^2/(1-4*x).
%F G.f.: 1/(1 - Sum_{j>=1} (3*j-1)*x^j). - _Joerg Arndt_, Jul 06 2011
%F a(n) = 4*a(n-1) + (-1)^n*C(2,2-n).
%F a(n) = Sum_{k=0..n} A201780(n,k)*2^k. - _Philippe Deléham_, Dec 05 2011
%t Join[{1, 2}, 9*4^Range[0, 30]] (* _Jean-François Alcover_, Jul 21 2018 *)
%Y Cf. A000302 and A002001.
%Y Essentially the same as A002063.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, May 30 2000
%E New name from _Joerg Arndt_, Jul 06 2011
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