A266720 Binary representation of the middle column of the "Rule 59" elementary cellular automaton starting with a single ON (black) cell.
1, 10, 101, 1011, 10110, 101101, 1011010, 10110101, 101101010, 1011010101, 10110101010, 101101010101, 1011010101010, 10110101010101, 101101010101010, 1011010101010101, 10110101010101010, 101101010101010101, 1011010101010101010, 10110101010101010101
Offset: 0
Links
- Robert Price, Table of n, a(n) for n = 0..1000
- 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
- Index entries for linear recurrences with constant coefficients, signature (10,1,-10).
Programs
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Mathematica
rule=59; 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 *) mc=Table[catri[[k]][[k]],{k,1,rows}]; (* Keep only middle cell from each row *) Table[FromDigits[Take[mc,k]],{k,1,rows}] (* Binary Representation of Middle Column *) -
Python
print([10009*10**n//9900 for n in range(50)]) # Karl V. Keller, Jr., Oct 18 2021
Formula
From Colin Barker, Jan 04 2016 and Apr 18 2019: (Start)
a(n) = (-450*(-1)^n+10009*10^n-550)/9900 for n>1.
a(n) = 10*a(n-1)+a(n-2)-10*a(n-3) for n>4.
G.f.: (1+x^3-x^4) / ((1-x)*(1+x)*(1-10*x)).
(End)
a(n) = floor(10009*10^n/9900). - Karl V. Keller, Jr., Oct 17 2021