cp's OEIS Frontend

A292686 Sierpinski-type iteration: start with a(0)=1, at each step, replace 0 with 000 and 1 with 101.

%I A292686 #25 May 19 2023 04:54:57
%S A292686 1,101,101000101,101000101000000000101000101,
%T A292686 101000101000000000101000101000000000000000000000000000101000101000000000101000101
%N A292686 Sierpinski-type iteration: start with a(0)=1, at each step, replace 0 with 000 and 1 with 101.
%C A292686 See A292687 for the decimal representation of a(n) viewed as a "binary number", i.e., as written in base 2.
%C A292686 The Sierpinski carpet (A153490) can be seen as 2-dimensional version of this 1-dimensional variant. The classical Sierpinski gasket triangle (Pascal's triangle mod 2) and "Rule 18" (or Rule 90, A070886) and "Rule 22" (A071029) have similar graphs.
%C A292686 The n-th term a(n) has 3^n digits, the middle third of which are all zero. The digits of a(n) are again the first and last 3^n digits of a(n+1), separated by 3^n zeros.
%H A292686 Michael Coons and James Evans, <a href="https://arxiv.org/abs/2011.10722">A sequential view of self--similar measures, or, What the ghosts of Mahler and Cantor can teach us about dimension</a>, arXiv:2011.10722 [math.NT], 2020. See Figure 2 p. 2.
%F A292686 a(n+1) = convert(5*a(n), from base 8, to base 2).
%F A292686 a(n+1) = (100^(3^n)+1)*a(n).
%F A292686 a(n) = Product_{k=0 .. n-1} (100^(3^k)+1).
%e A292686 a(0) = 1 -> 101 = a(1);
%e A292686 a(1) = 101 -> concat(101,000,101) = 101000101 = a(2).
%t A292686 A292686[nmax_]:=FoldList[Times,1,100^(3^Range[0,nmax-1])+1];A292686[5] (* _Paolo Xausa_, May 13 2023 *)
%o A292686 (PARI) a(n,a=1)=for(k=1,n,a=fromdigits(binary(a)*5,8));fromdigits(binary(a),10) \\ Illustration of the first formula.
%o A292686 (PARI) A292686(n)=prod(k=0,n-1,100^(3^k)+1)
%Y A292686 Cf. A292687 for the decimal representation of a(n) viewed as a "binary number", i.e., as written in base 2.
%Y A292686 Cf. A153490 (Sierpinski carpet), A047999 (Sierpinski gasket = Pascal's triangle mod 2), A070886 (Rule 18 / Rule 90), A071029 (Rule 22).
%Y A292686 Cf. A088917.
%K A292686 nonn
%O A292686 0,2
%A A292686 _M. F. Hasler_, Oct 20 2017