A267260 Total number of ON (black) cells after n iterations of the "Rule 111" elementary cellular automaton starting with a single ON (black) cell.

1, 3, 6, 11, 16, 23, 29, 40, 46, 61, 67, 86, 92, 115, 121, 148, 154, 185, 191, 226, 232, 271, 277, 320, 326, 373, 379, 430, 436, 491, 497, 556, 562, 625, 631, 698, 704, 775, 781, 856, 862, 941, 947, 1030, 1036, 1123, 1129, 1220, 1226, 1321, 1327, 1426, 1432

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 13 2016 and Apr 19 2019: (Start)

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

G.f.: (1+2*x+x^2+x^3-x^5-x^6+2*x^7-x^8) / ((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 *) Table[Total[Take[nbc, k]], {k, 1, rows}] (* Number of Black cells through stage n *)

Crossrefs

Cf. A267253.

Sequence in context: A024401 A333709 A266252 * A116940 A278100 A087099

Adjacent sequences: A267257 A267258 A267259 * A267261 A267262 A267263

Keyword

nonn,easy

Author

Robert Price, Jan 12 2016

Status

approved