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a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-4), with a(0)=a(1)=a(2)=0, and a(3)=1.
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%I #20 Sep 08 2022 08:45:32

%S 0,0,0,1,4,12,32,82,208,528,1344,3428,8752,22352,57088,145800,372352,

%T 950912,2428416,6201616,15837504,40445376,103288320,263775008,

%U 673621760,1720277760,4393200640,11219241536,28651407104,73169217792,186857644032,477192188032

%N a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-4), with a(0)=a(1)=a(2)=0, and a(3)=1.

%C The inverse binomial transform is {0, 0, 0, 1, 0, 2, 0, 5, 0, 12, 0, 29, ...} (n>=0), an aerated variant of A000129. - _R. J. Mathar_, Jul 10 2019

%H Colin Barker, <a href="/A135248/b135248.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 (4,-4,0,2).

%F G.f.: x^3 / (1-4*x+4*x^2-2*x^4). - _Colin Barker_, Apr 08 2016

%p seq(coeff(series(x^3/(1-4*x+4*x^2-2*x^4), x, n+1), x, n), n = 0 ..35); # _G. C. Greubel_, Nov 21 2019

%t LinearRecurrence[{4,-4,0,2}, {0,0,0,1}, 35] (* _G. C. Greubel_, Oct 04 2016 *)

%o (PARI) concat(vector(3), Vec(x^3/(1-4*x+4*x^2-2*x^4) + O(x^35))) \\ _Colin Barker_, Apr 08 2016

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x^3/(1-4*x+4*x^2-2*x^4) )); // _G. C. Greubel_, Nov 21 2019

%o (Sage)

%o def A135248_list(prec):

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

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

%o A135248_list(30) # _G. C. Greubel_, Nov 21 2019

%o (GAP) a:=[0,0,0,1];; for n in [5..35] do a[n]:=4*a[n-1]-4*a[n-2]+ 2*a[n-4]; od; a; # _G. C. Greubel_, Nov 21 2019

%Y Cf. A101893 (first differences).

%K nonn,easy

%O 0,5

%A _Paul Curtz_, Feb 15 2008