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A077958 Expansion of 1/(1-2*x^3).

%I A077958 #52 Mar 30 2025 02:44:54
%S A077958 1,0,0,2,0,0,4,0,0,8,0,0,16,0,0,32,0,0,64,0,0,128,0,0,256,0,0,512,0,0,
%T A077958 1024,0,0,2048,0,0,4096,0,0,8192,0,0,16384,0,0,32768,0,0,65536,0,0,
%U A077958 131072,0,0,262144,0,0,524288,0,0,1048576,0,0,2097152,0,0,4194304,0,0,8388608,0
%N A077958 Expansion of 1/(1-2*x^3).
%C A077958 a(n) is the number of L-tromino tilings of the n X 2 rectangle (see Exercise 2 in Grinberg). - _Stefano Spezia_, Nov 26 2019
%H A077958 G. C. Greubel, <a href="/A077958/b077958.txt">Table of n, a(n) for n = 0..1002</a>
%H A077958 Darij Grinberg, <a href="http://www.cip.ifi.lmu.de/~grinberg/t/19fco/hw1s.pdf">Math 222: Enumerative Combinatorics, Fall 2019: Homework 1</a>, Drexel University, Department of Mathematics, 2019.
%H A077958 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2).
%F A077958 From _Stefano Spezia_, Nov 26 2019: (Start)
%F A077958 a(n) = 2^(n/3) if 3 divides n, otherwise a(n) = 0 (see Exercise 2 in Grinberg).
%F A077958 E.g.f.: (1/3)*(exp(-(-2)^(1/3)*x) + exp(2^(1/3)*x) + exp((-1)^(2/3)*2^(1/3)*x)). (End)
%t A077958 CoefficientList[Series[1/(1-2*x^3),{x,0,80}],x] (* _Vladimir Joseph Stephan Orlovsky_, Jan 30 2012 *)
%t A077958 Riffle[Riffle[2^Range[0,30],0],0,{3,-1,3}] (* _Harvey P. Dale_, Dec 18 2012 *)
%o A077958 (PARI) Vec(1/(1-2*x^3)+O(x^80)) \\ _Charles R Greathouse IV_, Sep 27 2012
%o A077958 (PARI) a(n)= !(n%3)<<(n\3) \\ _Ruud H.G. van Tol_, Mar 28 2025
%o A077958 (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/(1-2*x^3) )); // _G. C. Greubel_, Jun 23 2019
%o A077958 (Sage) (1/(1-2*x^3)).series(x, 80).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 23 2019
%Y A077958 Cf. A006801, A162673, A163433, A329185.
%K A077958 nonn,easy
%O A077958 0,4
%A A077958 _N. J. A. Sloane_, Nov 17 2002