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

1, 0, 3, 0, 9, 0, 27, 0, 81, 0, 243, 0, 729, 0, 2187, 0, 6561, 0, 19683, 0, 59049, 0, 177147, 0, 531441, 0, 1594323, 0, 4782969, 0, 14348907, 0, 43046721, 0, 129140163, 0, 387420489, 0, 1162261467, 0, 3486784401, 0, 10460353203, 0, 31381059609, 0, 94143178827

Offset: 0

Author

Kival Ngaokrajang, Jan 26 2015

Keywords

• nonn
• easy

Comments

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

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.

Powers of 3 interspersed with zeros. - Colin Barker, Jan 26 2015

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Kival Ngaokrajang, Illustration of construction rule and initial terms
• Index entries for linear recurrences with constant coefficients, signature (0,3).

Crossrefs

Cf. A000244, A251732, A251733.

Programs

Mathematica

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 *)

PARI

{
a=1; print1(a,", ");
for (n=1,100,
     if (Mod(n,2)==0,
         a=a*3;
         print1(a,", "),
         print1(0,", ")
     )
)
}

PARI

Vec(-1/(3*x^2-1) + O(x^100)) \\ Colin Barker, Jan 26 2015

PARI

a(n) = if(n%2==0,3^(n/2),0) \\ Jason Yuen, Mar 24 2025

Formula

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

a(n) = (3^(n/2)*(1+(-1)^n))/2. - Colin Barker, Jan 26 2015

G.f.: -1 / (3*x^2-1). - Colin Barker, Jan 26 2015