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A014236 Expansion of g.f.: 2*x*(1-x)/((1-2*x)*(1-2*x^2)).

%I A014236 #38 Aug 15 2025 07:35:44
%S A014236 0,2,2,8,12,32,56,128,240,512,992,2048,4032,8192,16256,32768,65280,
%T A014236 131072,261632,524288,1047552,2097152,4192256,8388608,16773120,
%U A014236 33554432,67100672,134217728,268419072,536870912,1073709056,2147483648,4294901760,8589934592
%N A014236 Expansion of g.f.: 2*x*(1-x)/((1-2*x)*(1-2*x^2)).
%C A014236 Number of symmetric chiral (optically active) isomers possible for organic compounds with n distinct carbon atoms.
%C A014236 A020522 interleaved with A004171 and apparently the number of asymmetric Dyck (n+2)-paths with exactly half of the steps lying between the first and last peaks; e.g. all asymmetric 3-paths (UU*DDU*D and U*DUU*DD) comply so a(1)=2. - _David Scambler_, Sep 14 2012
%H A014236 Harvey P. Dale, <a href="/A014236/b014236.txt">Table of n, a(n) for n = 0..1000</a>
%H A014236 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-4).
%F A014236 a(n) = 2*A007179(n). - _R. J. Mathar_, Nov 14 2011
%F A014236 From _G. C. Greubel_, Jun 22 2019: (Start)
%F A014236 a(n) = 2^((n - 2)/2)*(2^((n + 2)/2) - 1 - (-1)^n).
%F A014236 E.g.f.: exp(2*x) - cosh(sqrt(2)*x). (End)
%p A014236 f := n -> if n mod 2 = 0 then 2^n-2^(n/2) else 2^n; fi;
%t A014236 CoefficientList[Series[2x (1-x)/((1-2x)(1-2x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{2,2,-4},{0,2,2},30] (* _Harvey P. Dale_, Dec 04 2018 *)
%o A014236 (PARI) my(x='x+O('x^30)); concat([0], Vec(2*x*(1-x)/((1-2*x)*(1-2*x^2)))) \\ _G. C. Greubel_, Jun 22 2019
%o A014236 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( 2*x*(1-x)/((1-2*x)*(1-2*x^2)) )); // _G. C. Greubel_, Jun 22 2019
%o A014236 (Sage) (2*x*(1-x)/((1-2*x)*(1-2*x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 22 2019
%o A014236 (GAP) a:=[0,2,2];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-4*a[n-3]; od; a; # _G. C. Greubel_, Jun 22 2019
%Y A014236 Second differences of A027556.
%K A014236 nonn,easy
%O A014236 0,2
%A A014236 Paul F. Hudrlik (hudrlik(AT)scs.howard.edu)
%E A014236 G.f. corrected by _Olivier Gérard_, Nov 13 2011