A267460 Number of OFF (white) cells in the n-th iteration of the "Rule 133" elementary cellular automaton starting with a single ON (black) cell.

0, 2, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 46, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 58, 60, 60, 62, 62, 64, 64, 66, 66, 68

Offset: 0

Author

Robert Price, Jan 15 2016

Keywords

• nonn

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. A267423.

Cf. A052928, A265283, A213222.

Programs

Mathematica

rule=133; 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 *) nbc=Table[Total[catri[[k]]],{k,1,rows}]; (* Number of Black cells in stage n *) Table[Length[catri[[k]]]-nbc[[k]],{k,1,rows}] (* Number of White cells in stage n *)

Formula

Conjectures from Colin Barker, Jan 16 2016: (Start)

a(n) = (2*n-(-1)^n+5)/2 for n>1.

a(n) = a(n-1)+a(n-2)-a(n-3) for n>4.

G.f.: 2*x*(1+x-x^3) / ((1-x)^2*(1+x)).

(End)