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A099787 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * 2^k * 3^(n-4*k).

%I A099787 #11 Sep 08 2022 08:45:15
%S A099787 1,3,9,27,83,267,909,3267,12235,46983,182529,711099,2764619,10704147,
%T A099787 41257341,158371011,605932099,2312728095,8812918161,33549513579,
%U A099787 127652354627,485608571547,1847326271949,7028217617859,26742885359131
%N A099787 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * 2^k * 3^(n-4*k).
%C A099787 In general a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * u^k * v^(n-4*k) has g.f. (1-v*x)^2/((1-v*x)^3 - u*x^4) and satisfies the recurrence a(n) =  3*v*a(n-1) - 3*v^2*a(n-2) + v^3*a(n-3) + u*a(n-4).
%H A099787 G. C. Greubel, <a href="/A099787/b099787.txt">Table of n, a(n) for n = 0..1000</a>
%H A099787 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (9,-27,27,2).
%F A099787 G.f.: (1-3*x)^2/((1-3*x)^3 - 2*x^4).
%F A099787 a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3) + 2*a(n-4).
%p A099787 seq(coeff(series((1-3*x)^2/((1-3*x)^3 - 2*x^4), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Sep 04 2019
%t A099787 LinearRecurrence[{9,-27,27,2}, {1,3,9,27}, 30] (* _G. C. Greubel_, Sep 04 2019 *)
%o A099787 (PARI) my(x='x+O('x^30)); Vec((1-3*x)^2/((1-3*x)^3 - 2*x^4)) \\ _G. C. Greubel_, Sep 04 2019
%o A099787 (Magma) I:=[1,3,9,27]; [n le 4 select I[n] else 9*Self(n-1) - 27*Self(n-2) + 27*Self(n-3) + 2*Self(n-4): n in [1..30]]; // _G. C. Greubel_, Sep 04 2019
%o A099787 (Sage)
%o A099787 def A099787_list(prec):
%o A099787     P.<x> = PowerSeriesRing(ZZ, prec)
%o A099787     return P((1-3*x)^2/((1-3*x)^3 - 2*x^4)).list()
%o A099787 A099787_list(30) # _G. C. Greubel_, Sep 04 2019
%o A099787 (GAP) a:=[1,3,9,27];; for n in [5..30] do a[n]:=9*a[n-1]-27*a[n-2] + 27*a[n-3] +2*a[n-4]; od; a; # _G. C. Greubel_, Sep 04 2019
%Y A099787 Cf. A099780, A099781, A099783, A099784, A099785, A099786.
%K A099787 easy,nonn
%O A099787 0,2
%A A099787 _Paul Barry_, Oct 26 2004