Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #22 Jan 23 2019 02:32:31
%S 1,4,16,64,229,592,-224,-18176,-175655,-1265732,-7914560,-44970752,
%T -236014307,-1145932664,-5086940240,-19929220352,-61944816911,
%U -81359219468,858917862064,10785877546432,84667993188757,555461238134080,3268576565244544,17688312222825472,88631554966652233,408731119650234796
%N a(n) = Sum_{k=0..floor(n/4)} binomial(n-k, 3k)*(-3)^(3k)*4^(n-4k).
%H P. S. Bruckman and G. C. Greubel, <a href="https://www.fq.math.ca/Problems/MayAdvanced2014.pdf">Advanced Problem H-725</a>, Fibonacci Quarterly, 52(2):187-190, 2014.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (12,-48,64,-27).
%F a(n) = (9n + 7 + 3^(3n/2)*(11*sqrt(2)*cos(n*arcsin(sqrt(2/27))) + sin(n*arcsin(sqrt(2/27))))/sqrt(2))/18.
%F G.f.: (1-4x)^2/((1-4x)^3+27x^4) = (1-4*x)^2/((x-1)^2*(1 - 10*x + 27*x^2)).
%p Gser:=series((1-4*x)^2/((1-4*x)^3+27*x^4), x = 0,35): seq(coeff(Gser, x, n), n = 0 .. 30);
%t LinearRecurrence[{12,-48,64,-27},{1,4,16,64},30] (* _Harvey P. Dale_, Nov 21 2015 *)
%K sign,easy
%O 0,2
%A _Emeric Deutsch_, Oct 27 2014