A266725 Total number of OFF (white) cells after n iterations of the "Rule 59" elementary cellular automaton starting with a single ON (black) cell.

0, 1, 4, 5, 12, 13, 24, 25, 40, 41, 60, 61, 84, 85, 112, 113, 144, 145, 180, 181, 220, 221, 264, 265, 312, 313, 364, 365, 420, 421, 480, 481, 544, 545, 612, 613, 684, 685, 760, 761, 840, 841, 924, 925, 1012, 1013, 1104, 1105, 1200, 1201, 1300, 1301, 1404

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 05 2016 and Apr 18 2019: (Start)

a(n) = (2*n*(n+(-1)^n+1)-(-1)^n+1)/4.

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>4.

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

(End)

Mathematica

rule=59; 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 *) nwc=Table[Length[catri[[k]]]-nbc[[k]], {k, 1, rows}]; (* Number of White cells in stage n *) Table[Total[Take[nwc, k]], {k, 1, rows}] (* Number of White cells through stage n *)

Crossrefs

Cf. A266716.

Sequence in context: A325688 A080277 A047608 * A308783 A130011 A050022

Adjacent sequences: A266722 A266723 A266724 * A266726 A266727 A266728

Keyword

nonn

Author

Robert Price, Jan 03 2016

Status

approved