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A266976 Decimal representation of the n-th iteration of the "Rule 78" elementary cellular automaton starting with a single ON (black) cell.

%I A266976 #29 Feb 16 2025 08:33:29
%S A266976 1,6,28,104,464,1696,7488,27264,120064,436736,1922048,6989824,
%T A266976 30756864,111845376,492126208,1789558784,7874084864,28633071616,
%U A266976 125985619968,458129670144,2015770968064,7330076819456,32252339683328,117281237499904,516037451710464
%N A266976 Decimal representation of the n-th iteration of the "Rule 78" elementary cellular automaton starting with a single ON (black) cell.
%H A266976 Robert Price, <a href="/A266976/b266976.txt">Table of n, a(n) for n = 0..1000</a>
%H A266976 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A266976 Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>, Wolfram Media, 2002; p. 55.
%H A266976 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A266976 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A266976 Conjectures from _Colin Barker_, Jan 08 2016 and Apr 18 2019: (Start)
%F A266976 a(n) = 2^(n-2)*((-2)^n+21*2^n-4)/3 = 2^(n-1)*A277954(n+1) for n>0.
%F A266976 a(n) = 2*a(n-1) + 16*a(n-2) - 32*a(n-3) for n>3.
%F A266976 G.f.: (1+4*x-16*x^3) / ((1-2*x)*(1-4*x)*(1+4*x)).
%F A266976 (End)
%t A266976 rule=78; 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 A266976 Cf. A266974, A266975.
%K A266976 nonn,easy
%O A266976 0,2
%A A266976 _Robert Price_, Jan 07 2016
%E A266976 Removed an unjustified claim that _Colin Barker_'s conjectures are correct. Removed a program based on a conjecture. - _Michael De Vlieger_, Jun 13 2022