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

1, 101, 101000101, 101000101000000000101000101, 101000101000000000101000101000000000000000000000000000101000101000000000101000101

Offset

0,2

Comments

See A292687 for the decimal representation of a(n) viewed as a "binary number", i.e., as written in base 2.

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.

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.

Links

Table of n, a(n) for n=0..4.

Michael Coons and James Evans, A sequential view of self--similar measures, or, What the ghosts of Mahler and Cantor can teach us about dimension, arXiv:2011.10722 [math.NT], 2020. See Figure 2 p. 2.

Formula

a(n+1) = convert(5*a(n), from base 8, to base 2).

a(n+1) = (100^(3^n)+1)*a(n).

a(n) = Product_{k=0 .. n-1} (100^(3^k)+1).

Example

a(0) = 1 -> 101 = a(1);
a(1) = 101 -> concat(101,000,101) = 101000101 = a(2).

Mathematica

A292686[nmax_]:=FoldList[Times, 1, 100^(3^Range[0, nmax-1])+1]; A292686[5] (* Paolo Xausa, May 13 2023 *)

Prog

(PARI) a(n, a=1)=for(k=1, n, a=fromdigits(binary(a)*5, 8)); fromdigits(binary(a), 10) \\ Illustration of the first formula.
(PARI) A292686(n)=prod(k=0, n-1, 100^(3^k)+1)

Crossrefs

Cf. A292687 for the decimal representation of a(n) viewed as a "binary number", i.e., as written in base 2.

Cf. A153490 (Sierpinski carpet), A047999 (Sierpinski gasket = Pascal's triangle mod 2), A070886 (Rule 18 / Rule 90), A071029 (Rule 22).

Cf. A088917.

Sequence in context: A365994 A262645 A138826 * A138720 A262627 A259199

Adjacent sequences: A292683 A292684 A292685 * A292687 A292688 A292689

Keyword

nonn

Author

M. F. Hasler, Oct 20 2017

Status

approved