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A292686 Sierpinski-type iteration: start with a(0)=1, at each step, replace 0 by 000 and 1 by 101. 3
1, 101, 101000101, 101000101000000000101000101, 101000101000000000101000101000000000000000000000000000101000101000000000101000101 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

See A292687 for the decimal representation of a(n) viewed as "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.

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).

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 "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).

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

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Last modified March 19 00:15 EDT 2019. Contains 321306 sequences. (Running on oeis4.)