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A266380 Decimal representation of the n-th iteration of the "Rule 21" elementary cellular automaton starting with a single ON (black) cell.

%I A266380 #32 May 06 2025 08:35:22
%S A266380 1,3,0,127,0,2047,0,32767,0,524287,0,8388607,0,134217727,0,2147483647,
%T A266380 0,34359738367,0,549755813887,0,8796093022207,0,140737488355327,0,
%U A266380 2251799813685247,0,36028797018963967,0,576460752303423487,0,9223372036854775807,0
%N A266380 Decimal representation of the n-th iteration of the "Rule 21" elementary cellular automaton starting with a single ON (black) cell.
%D A266380 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A266380 Robert Price, <a href="/A266380/b266380.txt">Table of n, a(n) for n = 0..500</a>
%H A266380 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A266380 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A266380 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A266380 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,17,0,-16).
%F A266380 From _Colin Barker_, Dec 29 2015 and Apr 15 2019: (Start)
%F A266380 a(n) = 17*a(n-2) - 16*a(n-4) for n>5.
%F A266380 G.f.: (1 + 3*x - 17*x^2 + 76*x^3 + 16*x^4 - 64*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)). (End)
%F A266380 a(n) = (1 - (-1)^n)*(4*16^floor(n/2) - 1/2) for n>1. - _Bruno Berselli_, Dec 29 2015
%F A266380 a(n) = (2*4^n - 1)*(n mod 2) + 0^n - 4*0^abs(n-1). - _Karl V. Keller, Jr._, Sep 03 2021
%t A266380 rule=21; 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 A266380 (Magma) [n le 1 select 3^n else (1-(-1)^n)*(4*16^Floor(n/2)-1/2): n in [0..40]]; // _Bruno Berselli_, Dec 29 2015
%o A266380 (Python) print([(2*4**n - 1)*(n%2) + 0**n - 4*0**abs(n-1) for n in range(50)]) # _Karl V. Keller, Jr._, Sep 03 2021
%Y A266380 Cf. A266377, A266379.
%Y A266380 Cf. A241955: a(2*n+1) for n>0.
%Y A266380 Essentially the same as A266324 and A266218.
%K A266380 nonn,easy
%O A266380 0,2
%A A266380 _Robert Price_, Dec 28 2015