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Decimal representation of the n-th iteration of the "Rule 103" elementary cellular automaton starting with a single ON (black) cell.
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%I #35 Feb 16 2025 08:33:29

%S 1,6,6,117,60,1931,376,32535,752,523823,1504,8387679,3008,134215871,

%T 6016,2147479935,12032,34359730943,24064,549755799039,48128,

%U 8796092992511,96256,140737488295935,192512,2251799813566463,385024,36028797018726399,770048

%N Decimal representation of the n-th iteration of the "Rule 103" elementary cellular automaton starting with a single ON (black) cell.

%H Robert Price, <a href="/A267139/b267139.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 11 2016 and Apr 19 2019: (Start)

%F a(n) = 19*a(n-2) - 50*a(n-4) + 32*a(n-6) for n > 10.

%F G.f.: (1 + 6*x - 13*x^2 + 3*x^3 - 4*x^4 + 8*x^5 - 496*x^6 + 1504*x^7 - 3584*x^8 - 1536*x^9 + 4096*x^10) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)). (End)

%t rule=103; 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. A267136, A267138.

%K nonn,easy,changed

%O 0,2

%A _Robert Price_, Jan 10 2016

%E Removed an unjustified claim that _Colin Barker_'s conjectures are correct. Removed a program based on a conjecture. - _Michael De Vlieger_, Jun 13 2022