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Binary representation of the n-th iteration of the "Rule 41" elementary cellular automaton starting with a single ON (black) cell.
2

%I #29 Feb 16 2025 08:33:28

%S 1,0,101,1100100,1,11111111000,101,111111111100100,1,

%T 1111111111111111000,101,11111111111111111100100,1,

%U 111111111111111111111111000,101,1111111111111111111111111100100,1,11111111111111111111111111111111000,101

%N Binary representation of the n-th iteration of the "Rule 41" elementary cellular automaton starting with a single ON (black) cell.

%H Robert Price, <a href="/A266609/b266609.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>, Wolfram Media, 2002; p. 55.

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%F Conjectures from _Colin Barker_, Jan 02 2016 and Apr 18 2019: (Start)

%F a(n) = 10000*a(n-2)+a(n-4)-10000*a(n-6) for n>5.

%F G.f.: (1-9899*x^2+1100100*x^3-1010000*x^4+110111000*x^5) / ((1-x)*(1+x)*(1-100*x)*(1+100*x)*(1+x^2)).

%F (End)

%F Conjecture: a(n) = floor((10*100^n + 10 - 1010*(n mod 4))/9) - 10675*0^((n+1) mod 4) for odd n; a(n) = 101^((n mod 4)/2) for even n. - _Karl V. Keller, Jr._, Oct 07 2021

%t rule=41; 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 *)

%Y Cf. A266608, A266610.

%K nonn,easy,changed

%O 0,3

%A _Robert Price_, Jan 01 2016