A266179 Binary representation of the n-th iteration of the "Rule 6" elementary cellular automaton starting with a single ON (black) cell.

1, 110, 10000, 1100000, 100000000, 11000000000, 1000000000000, 110000000000000, 10000000000000000, 1100000000000000000, 100000000000000000000, 11000000000000000000000, 1000000000000000000000000, 110000000000000000000000000, 10000000000000000000000000000

Offset: 0

Author

Robert Price, Dec 22 2015

Keywords

• nonn
• easy

Comments

Rules 38,134 and 166 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..999
• 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
• Index entries for linear recurrences with constant coefficients, signature (0, 10000).

Crossrefs

Cf. A266178, A266180.

Programs

Mathematica

rule=6; 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([110*100**(n-1) if n%2 else 100**n for n in range(30)]) # Karl V. Keller, Jr., Aug 11 2021

Formula

From Colin Barker, Dec 23 2015 and Apr 13 2019: (Start)

a(n) = 4^(n-1)*5^(2*n-1)*(21-(-1)^n).

a(n) = 10000*a(n-2) for n>1.

G.f.: (1+110*x) / ((1-100*x)*(1+100*x)). (End)

a(n) = 110*100^(n-1) for odd n; a(n) = 100^n for even n. - Karl V. Keller, Jr., Aug 11 2021