A254006 - OEIS

%I #34 Nov 30 2015 18:00:30

%S 1,0,3,0,9,0,27,0,81,0,243,0,729,0,2187,0,6561,0,19683,0,59049,0,

%T 177147,0,531441,0,1594323,0,4782969,0,14348907,0,43046721,0,

%U 129140163,0,387420489,0,1162261467,0,3486784401,0,10460353203,0,31381059609,0,94143178827

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

%C Inspired by the Lévy C-curve, and generated using different construction rules as shown in the links.

%C 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 Powers of 3 interspersed with zeros. - _Colin Barker_, Jan 26 2015

%H Colin Barker, <a href="/A254006/b254006.txt">Table of n, a(n) for n = 0..1000</a>

%H Kival Ngaokrajang, <a href="/A254006/a254006_1.pdf">Illustration of construction rule and initial terms</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,3).

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

%F a(n) = (3^(n/2)*(1+(-1)^n))/2. - _Colin Barker_, Jan 26 2015

%F G.f.: -1 / (3*x^2-1). - _Colin Barker_, Jan 26 2015

%t 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 (PARI)

%o {

%o a=1; print1(a,", ");

%o for (n=1,100,

%o      if (Mod(n,2)==0,

%o          a=a*3;

%o          print1(a,", "),

%o          print1(0,", ")

%o      )

%o )

%o }

%o (PARI)

%o Vec(-1/(3*x^2-1) + O(x^100)) \\ _Colin Barker_, Jan 26 2015

%Y Cf. A251732, A251733.

%K nonn,easy

%O 0,3

%A _Kival Ngaokrajang_, Jan 26 2015