A267458 Number of ON (black) cells in the n-th iteration of the "Rule 133" elementary cellular automaton starting with a single ON (black) cell.

1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25, 27, 27, 29, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 39, 41, 41, 43, 43, 45, 45, 47, 47, 49, 49, 51, 51, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 63, 63, 65, 65

Offset: 0

Author

Robert Price, Jan 15 2016

Keywords

• nonn
• easy

Comments

1,1, followed by A109613 (odd integers repeated).

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.

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 *) Table[Total[catri[[k]]],{k,1,rows}] (* Number of Black cells in stage n *)

Formula

a(n) = A109613(n-2) for n>=2.

Conjectures from Colin Barker, Jan 15 2016 and Apr 19 2019: (Start)

a(n) = (2*n+(-1)^n-3)/2 for n>1.

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

G.f.: (1-x^2+2*x^4) / ((1-x)^2*(1+x)).

(End)