A266752 Binary representation of the n-th iteration of the "Rule 163" elementary cellular automaton starting with a single ON (black) cell.

1, 100, 10010, 1001010, 100101010, 10010101010, 1001010101010, 100101010101010, 10010101010101010, 1001010101010101010, 100101010101010101010, 10010101010101010101010, 1001010101010101010101010, 100101010101010101010101010, 10010101010101010101010101010, 1001010101010101010101010101010

Offset: 0

Author

Robert Price, Jan 17 2016

Keywords

• nonn
• easy

References

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

Links

• Robert Price, Table of n, a(n) for n = 0..1000
• 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. A263919, A266753.

Programs

Mathematica

rule=163; 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 *)

Formula

Conjectures from Colin Barker, Jan 20 2016 and Apr 18 2019: (Start)

a(n) = 101*a(n-1) - 100*a(n-2) for n>2.

G.f.: (1-x+10*x^2) / ((1-x)*(1-100*x)). (End)

Conjectured e.g.f.: (99 - 100*exp(x) + 991*exp(100*x))/990. - Stefano Spezia, Aug 05 2024