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%I #26 Jul 30 2022 08:19:39
%S 1,1,1,0,1,1,1,0,0,1,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,
%T 0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,
%U 1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,1,1,1
%N Triangle read by rows giving successive states of cellular automaton generated by "Rule 206" initiated with a single ON (black) cell.
%C Row n has length 2n+1.
%C This sequence is also generated by Rule 238.
%H Robert Price, <a href="/A267708/b267708.txt">Table of n, a(n) for n = 0..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://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 Conjecture: a(n) = (1 + (-1)^floor(2*sqrt(n)))/2. - _Velin Yanev_, Nov 30 2016
%F a(n) = A351319(n) for n != 2. - _Chai Wah Wu_, Jul 29 2022
%e The first ten rows:
%e 1
%e 1 1 0
%e 1 1 1 0 0
%e 1 1 1 1 0 0 0
%e 1 1 1 1 1 0 0 0 0
%e 1 1 1 1 1 1 0 0 0 0 0
%e 1 1 1 1 1 1 1 0 0 0 0 0 0
%e 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0
%e 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
%e 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0
%t rule=206; 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 *) Flatten[catri] (* Triangle Representation of CA *)
%o (Python)
%o from math import isqrt
%o def A267708(n): return int((k:=isqrt(n))**2+k-n+1>0) # _Chai Wah Wu_, Jul 28 2022
%Y Cf. A109241, A171476, A351319.
%Y Cf. A028242 (run lengths).
%K nonn,tabf,easy
%O 0
%A _Robert Price_, Jan 19 2016
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