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%I #43 Sep 08 2022 08:46:15
%S 1,1,8,79,64,1663,512,29695,4096,499711,32768,8191999,262144,
%T 132644863,2097152,2134900735,16777216,34259075071,134217728,
%U 548950507519,1073741824,8789650571263,8589934592,140685948747775,68719476736,2251387496824831,549755813888
%N Decimal representation of the n-th iteration of the "Rule 17" elementary cellular automaton starting with a single ON (black) cell.
%D Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H G. C. Greubel, <a href="/A266090/b266090.txt">Table of n, a(n) for n = 0..500</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%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>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,25,0,-152,0,128).
%F From _Colin Barker_, Dec 29 2015 and Apr 15 2019: (Start)
%F a(n) = 25*a(n-2)-152*a(n-4)+128*a(n-6) for n>5.
%F G.f.: (1-2*x)*(1+3*x-11*x^2+32*x^3+80*x^4) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-8*x^2)). (End)
%F a(n) = 8^(n/2) + (1-(-1)^n)*(2*4^n-8^(n/2)-6*8^((n-1)/2)-1)/2. Therefore: for even n, a(n) = 8^(n/2); otherwise, a(n) = 2*4^n - 6*8^((n-1)/2) - 1. - _Bruno Berselli_, Dec 29 2015
%t rule=17; 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 (Sage) [8^(n/2)+(1-(-1)^n)*(2*4^n-8^(n/2)-6*8^((n-1)/2)-1)/2 for n in [0..30]] # _Bruno Berselli_, Dec 29 2015
%o (Magma) [IsEven(n) select 8^(n div 2) else 2*4^n-6*8^((n-1) div 2)-1: n in [0..30]]; // _Bruno Berselli_, Dec 29 2015
%o (Python) print([2*4**n - 6*8**((n-1)//2) - 1 if n%2 else 8**(n//2) for n in range(50)]) # _Karl V. Keller, Jr._, Aug 31 2021
%Y Cf. A260552, A260692.
%K nonn,easy
%O 0,3
%A _Robert Price_, Dec 27 2015