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%I #25 Jul 06 2023 13:03:51
%S 1,4,10,97,92,1803,736,30815,5888,508671,47104,8263679,376832,
%T 133218303,3014656,2139488255,24117248,34295775231,192937984,
%U 549244108799,1543503872,8791999381503,12348030976,140704739229695,98784247808,2251537820680191,790273982464
%N Decimal representation of the n-th iteration of the "Rule 67" elementary cellular automaton starting with a single ON (black) cell.
%H Robert Price, <a href="/A266839/b266839.txt">Table of n, a(n) for n = 0..1000</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 Conjectures from _Colin Barker_, Jan 05 2016 and Apr 18 2019: (Start)
%F a(n) = 25*a(n-2) - 152*a(n-4) + 128*a(n-6) for n > 9.
%F G.f.: (1 + 4*x - 15*x^2 - 3*x^3 - 6*x^4 - 14*x^5 - 172*x^6 - 28*x^7 + 192*x^8 - 64*x^9) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-8*x^2)).
%F (End)
%F Conjecture: a(n) = 2*4^n - 61*8^floor(n/2)/16 - 1 for odd n > 2; a(n) = 23*8^(n/2)/16 for even n > 3. - _Karl V. Keller, Jr._, Dec 16 2021
%t rule=67; 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]],2],{k,1,rows}] (* Decimal Representation of Rows *)
%Y Cf. A266837, A266838.
%K nonn,easy
%O 0,2
%A _Robert Price_, Jan 04 2016