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a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * 2^(n-3*k).
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%I #12 Sep 08 2022 08:45:15

%S 1,2,4,8,18,48,144,448,1380,4152,12224,35456,102024,292768,840416,

%T 2416384,6959504,20069280,57913536,167158656,482462752,1392319488,

%U 4017460224,11590946816,33439639616,96470796672,278311599616

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

%C 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 G. C. Greubel, <a href="/A099785/b099785.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8,2).

%F G.f.: (1-2*x)^2/((1-2*x)^3 - 2*x^4).

%F a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) + 2*a(n-4).

%p seq(coeff(series((1-2*x)^2/((1-2*x)^3 - 2*x^4), x, n+1), x, n), n = 0 .. 40); # _G. C. Greubel_, Sep 04 2019

%t Table[Sum[Binomial[n-k,3k]2^(n-3k),{k,0,Floor[n/4]}],{n,0,30}] (* or *) LinearRecurrence[{6,-12,8,2},{1,2,4,8},30] (* _Harvey P. Dale_, Apr 01 2012 *)

%o (PARI) my(x='x+O('x^30)); Vec((1-2*x)^2/((1-2*x)^3 - 2*x^4)) \\ _G. C. Greubel_, Sep 04 2019

%o (Magma) I:=[1,2,4,8]; [n le 4 select I[n] else 6*Self(n-1) - 12*Self(n-2) + 8*Self(n-3) + 2*Self(n-4): n in [1..30]]; // _G. C. Greubel_, Sep 04 2019

%o (Sage)

%o def A099785_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P((1-2*x)^2/((1-2*x)^3 - 2*x^4)).list()

%o A099785_list(30) # _G. C. Greubel_, Sep 04 2019

%o (GAP) a:=[1,2,4,8];; for n in [5..30] do a[n]:=6*a[n-1] -12*a[n-2] + 8*a[n-3] +2*a[n-4]; od; a; # _G. C. Greubel_, Sep 04 2019

%Y Cf. A003522, A097119.

%Y Cf. A099780, A099781, A099783, A099784, A099785, A099786, A099787.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Oct 26 2004