

A292686


Sierpinskitype iteration: start with a(0)=1, at each step, replace 0 with 000 and 1 with 101.


3




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 2dimensional version of this 1dimensional 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 nth 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 selfsimilar 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 .. n1} (100^(3^k)+1).


EXAMPLE

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


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, n1, 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: A082521 A262645 A138826 * A138720 A262627 A259199
Adjacent sequences: A292683 A292684 A292685 * A292687 A292688 A292689


KEYWORD

nonn


AUTHOR

M. F. Hasler, Oct 20 2017


STATUS

approved



