A266721 Decimal representation of the middle column of the "Rule 59" elementary cellular automaton starting with a single ON (black) cell.
1, 2, 5, 11, 22, 45, 90, 181, 362, 725, 1450, 2901, 5802, 11605, 23210, 46421, 92842, 185685, 371370, 742741, 1485482, 2970965, 5941930, 11883861, 23767722, 47535445, 95070890, 190141781, 380283562, 760567125, 1521134250, 3042268501, 6084537002, 12169074005
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 (2,1,-2).
Programs
-
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],2],{k,1,rows}] (* Binary Representation of Middle Column *) -
Python
print([17*2**n//12 for n in range(50)]) # Karl V. Keller, Jr., Oct 18 2021
Formula
From Colin Barker, Jan 05 2016 and Apr 17 2019: (Start)
a(n) = (-2*(-1)^n+17*2^n-6)/12 for n>1.
a(n) = 2*a(n-1)+a(n-2)-2*a(n-3) for n>4.
G.f.: (1+x^3-x^4) / ((1-x)*(1+x)*(1-2*x)). (End)
a(n) = floor(17*2^n/12). - Karl V. Keller, Jr., Oct 17 2021