A267262 Total number of OFF (white) cells after n iterations of the "Rule 111" elementary cellular automaton starting with a single ON (black) cell.

0, 1, 3, 5, 9, 13, 20, 24, 35, 39, 54, 58, 77, 81, 104, 108, 135, 139, 170, 174, 209, 213, 252, 256, 299, 303, 350, 354, 405, 409, 464, 468, 527, 531, 594, 598, 665, 669, 740, 744, 819, 823, 902, 906, 989, 993, 1080, 1084, 1175, 1179, 1274, 1278, 1377, 1381

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) = (n^2+(-1)^n*(n-4)+2)/2 for n>3.

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>8.

G.f.: x*(1+2*x+x^4+x^5-2*x^6+x^7) / ((1-x)^3*(1+x)^2).

(End)

Mathematica

rule=111; 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. A267253.

Sequence in context: A248956 A108754 A033499 * A106607 A305082 A007042

Adjacent sequences: A267259 A267260 A267261 * A267263 A267264 A267265

Keyword

nonn

Author

Robert Price, Jan 12 2016

Status

approved