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A254006 a(0) = 1, a(n) = 3*a(n-2) if n mod 2 = 0, otherwise a(n) = 0.

%I A254006 #42 Mar 24 2025 04:00:03
%S A254006 1,0,3,0,9,0,27,0,81,0,243,0,729,0,2187,0,6561,0,19683,0,59049,0,
%T A254006 177147,0,531441,0,1594323,0,4782969,0,14348907,0,43046721,0,
%U A254006 129140163,0,387420489,0,1162261467,0,3486784401,0,10460353203,0,31381059609,0,94143178827
%N A254006 a(0) = 1, a(n) = 3*a(n-2) if n mod 2 = 0, otherwise a(n) = 0.
%C A254006 Inspired by the Lévy C-curve, and generated using different construction rules as shown in the links.
%C A254006 The length of this variant Lévy C-curve is an integer in the real quadratic number field Q(sqrt(3)), namely L(n) = A(n) + B(n)*sqrt(3) with A(n) = a(n) and B(n) = a(n-1), with  a(0) = 1. See the construction rule and the illustration in the links.
%C A254006 Powers of 3 interspersed with zeros. - _Colin Barker_, Jan 26 2015
%H A254006 Colin Barker, <a href="/A254006/b254006.txt">Table of n, a(n) for n = 0..1000</a>
%H A254006 Kival Ngaokrajang, <a href="/A254006/a254006_1.pdf">Illustration of construction rule and initial terms</a>
%H A254006 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,3).
%F A254006 a(n) = 3*a(n-2) if n mod 2 = 0, otherwise a(n) = 0, a(0) = 1.
%F A254006 a(n) = (3^(n/2)*(1+(-1)^n))/2. - _Colin Barker_, Jan 26 2015
%F A254006 G.f.: -1 / (3*x^2-1). - _Colin Barker_, Jan 26 2015
%t A254006 nxt[{n_,a_,b_}]:={n+1,b,If[OddQ[n],3a,0]}; Transpose[NestList[nxt,{1,1,0},50]][[2]] (* or *) With[{nn=25},Riffle[3^Range[0,nn],0]] (* _Harvey P. Dale_, Nov 30 2015 *)
%o A254006 (PARI)
%o A254006 {
%o A254006 a=1; print1(a,", ");
%o A254006 for (n=1,100,
%o A254006      if (Mod(n,2)==0,
%o A254006          a=a*3;
%o A254006          print1(a,", "),
%o A254006          print1(0,", ")
%o A254006      )
%o A254006 )
%o A254006 }
%o A254006 (PARI)
%o A254006 Vec(-1/(3*x^2-1) + O(x^100)) \\ _Colin Barker_, Jan 26 2015
%o A254006 (PARI)
%o A254006 a(n) = if(n%2==0,3^(n/2),0) \\ _Jason Yuen_, Mar 24 2025
%Y A254006 Cf. A000244, A251732, A251733.
%K A254006 nonn,easy
%O A254006 0,3
%A A254006 _Kival Ngaokrajang_, Jan 26 2015