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Decimal representation of the n-th iteration of the "Rule 77" elementary cellular automaton starting with a single ON (black) cell.
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%I #28 Jul 06 2023 13:07:39

%S 1,2,21,42,341,682,5461,10922,87381,174762,1398101,2796202,22369621,

%T 44739242,357913941,715827882,5726623061,11453246122,91625968981,

%U 183251937962,1466015503701,2932031007402,23456248059221,46912496118442,375299968947541,750599937895082

%N Decimal representation of the n-th iteration of the "Rule 77" elementary cellular automaton starting with a single ON (black) cell.

%H Robert Price, <a href="/A266873/b266873.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) = ((-1)^n + 3*2^(2*n+1) + (-1)^n*2^(2*n+1) - 3)/6.

%F a(n) = 17*a(n-2) - 16*a(n-4) for n>3.

%F G.f.: (1+2*x)*(1+4*x^2) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)).

%F (End)

%t rule=77; 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 *)

%o (PARI) a(n) = 4<<(n<<1 - bitand(n,1)) \ 3; \\ _Kevin Ryde_, Mar 10 2022

%Y Cf. A266872.

%K nonn,easy

%O 0,2

%A _Robert Price_, Jan 04 2016