A266435 Binary representation of the n-th iteration of the "Rule 23" elementary cellular automaton starting with a single ON (black) cell.

1, 111, 0, 1111111, 0, 11111111111, 0, 111111111111111, 0, 1111111111111111111, 0, 11111111111111111111111, 0, 111111111111111111111111111, 0, 1111111111111111111111111111111, 0, 11111111111111111111111111111111111, 0, 111111111111111111111111111111111111111

Offset: 0

Author

Robert Price, Dec 29 2015

Keywords

• nonn
• easy

Comments

Rules 31, 55, 63, 87, 95, 119 and 127 also generate this sequence.

References

• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

• Robert Price, Table of n, a(n) for n = 0..500
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• S. Wolfram, A New Kind of Science
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A266434, A266436.

Essentially the same as A266217.

Programs

Mathematica

rule=23; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]]],{k,1,rows}]   (* Binary Representation of Rows *)

Python

print([(10*100**n-1)//9*(n%2) + 0**n for n in range(33)]) # Karl V. Keller, Jr., Aug 11 2021

Formula

From Colin Barker, Dec 30 2015 and Apr 15 2019: (Start)

a(n) = ((-1)^n-1)*(1-10^(2*n+1))/18 for n>0.

a(n) = 10001*a(n-2)-10000*a(n-4) for n>4.

G.f.: (1+111*x-10001*x^2+1000*x^3+10000*x^4) / ((1-x)*(1+x)*(1-100*x)*(1+100*x)).

(End)

a(n) = (10*100^n-1)/9*(n mod 2) + 0^n. - Karl V. Keller, Jr., Aug 11 2021