A267461 Total number of OFF (white) cells after n iterations of the "Rule 133" elementary cellular automaton starting with a single ON (black) cell.

0, 2, 6, 12, 18, 26, 34, 44, 54, 66, 78, 92, 106, 122, 138, 156, 174, 194, 214, 236, 258, 282, 306, 332, 358, 386, 414, 444, 474, 506, 538, 572, 606, 642, 678, 716, 754, 794, 834, 876, 918, 962, 1006, 1052, 1098, 1146, 1194, 1244, 1294, 1346, 1398, 1452

Offset

0,2

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 16 2016: (Start)

a(n) = (2*n^2+12*n-(-1)^n-7)/4 for n>0.

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

G.f.: 2*x*(1+x-x^3) / ((1-x)^3*(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 *) 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. A267423.

Sequence in context: A032371 A162802 A330320 * A108585 A294062 A193764

Adjacent sequences: A267458 A267459 A267460 * A267462 A267463 A267464

Keyword

nonn

Author

Robert Price, Jan 15 2016

Status

approved