A118109 Binary representation of n-th iteration of the Rule 54 elementary cellular automaton starting with a single black cell.
1, 111, 10001, 1110111, 100010001, 11101110111, 1000100010001, 111011101110111, 10001000100010001, 1110111011101110111, 100010001000100010001, 11101110111011101110111, 1000100010001000100010001, 111011101110111011101110111, 10001000100010001000100010001, 1110111011101110111011101110111
Offset: 0
Examples
From _Michael De Vlieger_, Oct 07 2015: (Start)
First 8 rows, representing ON cells as "1", OFF cells within the bounds of ON cells as "0", interpreted as a binary number at left, the decimal equivalent appearing at right (A118108):
1 = 1
111 = 7
1 0001 = 17
111 0111 = 119
1 0001 0001 = 273
111 0111 0111 = 1911
1 0001 0001 0001 = 4369
111 0111 0111 0111 = 30583
10001 0001 0001 0001 = 69905
(End)
Links
- Robert Price, Table of n, a(n) for n = 0..499
- Eric Weisstein's World of Mathematics, Rule 54
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
- Index entries for sequences related to cellular automata
- Index to Elementary Cellular Automata
Crossrefs
Programs
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Mathematica
rule=54; 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]]], {k, 1, rows}] (* Binary Representation of Rows *) (* Robert Price, Feb 21 2016 *)
Formula
Conjectures from Colin Barker, Dec 08 2015 and Apr 16 2019: (Start)
a(n) = 10001*a(n-2)-10000*a(n-4) for n>3.
G.f.: (1+111*x) / ((1-x)*(1+x)*(1-100*x)*(1+100*x)).
(End)
Conjecture: a(n) = floor((10000+1100*(n mod 2))*100^n/9999). - Karl V. Keller, Jr., Sep 24 2021
Extensions
Terms changed to match definition, as suggested by Michael De Vlieger. - N. J. A. Sloane, Oct 17 2015