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

0,5

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

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)

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 *)

Crossrefs

Cf. A267423.

Sequence in context: A293701 A296063 A127630 * A109613 A063196 A351744

Adjacent sequences: A267455 A267456 A267457 * A267459 A267460 A267461

Keyword

nonn,easy

Author

Robert Price, Jan 15 2016

Status

approved