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A267614
Decimal representation of the n-th iteration of the "Rule 185" elementary cellular automaton starting with a single ON (black) cell.
2
1, 1, 11, 47, 191, 767, 3071, 12287, 49151, 196607, 786431, 3145727, 12582911, 50331647, 201326591, 805306367, 3221225471, 12884901887, 51539607551, 206158430207, 824633720831, 3298534883327, 13194139533311, 52776558133247, 211106232532991, 844424930131967
OFFSET
0,3
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
FORMULA
Conjectures from Colin Barker, Jan 18 2016 and Apr 20 2019: (Start)
a(n) = 5*a(n-1) - 4*a(n-2) for n>3.
G.f.: (1-4*x+10*x^2-4*x^3) / ((1-x)*(1-4*x)).
(End)
Empirical a(n) = 3*4^(n-1)-1 for n>1. - Colin Barker, Nov 25 2016 and Apr 20 2019
MATHEMATICA
rule=185; 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 *) Table[FromDigits[catri[[k]], 2], {k, 1, rows}] (* Decimal Representation of Rows *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Jan 18 2016
EXTENSIONS
Removed an unjustified claim that Colin Barker's conjectures are correct. Removed a program based on a conjecture. - Michael De Vlieger, Jun 13 2022
STATUS
approved