A267354 Number of OFF (white) cells in the n-th iteration of the "Rule 123" elementary cellular automaton starting with a single ON (black) cell.

0, 1, 2, 1, 6, 1, 10, 1, 14, 1, 18, 1, 22, 1, 26, 1, 30, 1, 34, 1, 38, 1, 42, 1, 46, 1, 50, 1, 54, 1, 58, 1, 62, 1, 66, 1, 70, 1, 74, 1, 78, 1, 82, 1, 86, 1, 90, 1, 94, 1, 98, 1, 102, 1, 106, 1, 110, 1, 114, 1, 118, 1, 122, 1, 126, 1, 130, 1, 134, 1, 138, 1

Offset

0,3

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

Conjectures from Colin Barker, Jan 14 2016: (Start)

a(n) = ((2*(-1)^n+2)*n-3*(-1)^n-1)/2 for n>0.

a(n) = 2*a(n-2)-a(n-4) for n>4.

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

(End)

a(n) = A266303(n-1). - R. J. Mathar, Jan 17 2016

Mathematica

rule=123; 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 *)

Crossrefs

Cf. A267349.

Sequence in context: A302129 A191093 A266303 * A139625 A053785 A233809

Adjacent sequences: A267351 A267352 A267353 * A267355 A267356 A267357

Keyword

nonn,changed

Author

Robert Price, Jan 13 2016

Status

approved