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%I #25 Oct 04 2019 07:56:06
%S 1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,
%T 15,16,16,17,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,26,26,
%U 27,27,28,28,29,29,30,30,31,31,32,32,33,33,34,34,35,35
%N a(n) = round(n/2) + 1 = ceiling(n/2) + 1 = floor((n+1)/2) + 1.
%C Number of ON (black) cells in the n-th iteration of the "Rule 70" elementary cellular automaton starting with a single ON (black) cell.
%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H Robert Price, <a href="/A080513/b080513.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 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</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_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F From _Colin Barker_, Jan 14 2016: (Start)
%F a(n) = (2*n-(-1)^n+5)/4.
%F a(n) = a(n-1)+a(n-2)-a(n-3) for n>2.
%F G.f.: (1+x-x^2) / ((1-x)^2*(1+x)).
%F (End)
%F a(n) = 1 + A110654(n). - _Philippe Deléham_, Nov 23 2016
%F a(n) = A008619(n+1) = A110654(n+2) = A110654(n)+1 = A004526(n+3) = A140106(n+5); a(n+2) = a(n) + 1 for all n >= 0. - _M. F. Hasler_, Feb 14 2019
%F a(n) = a(n-1)*a(n-2) - Sum_{i=0..n-3} a(i). - _Marc Morgenegg_, Oct 04 2019
%t rule=70; 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[Total[catri[[k]]],{k,1,rows}] (* Number of Black cells in stage n *)
%o (PARI) a(n) = (2*n-(-1)^n+5)/4 \\ _Colin Barker_, Jan 14 2016
%o (PARI) Vec((1+x-x^2)/((1-x)^2*(1+x)) + O(x^100)) \\ _Colin Barker_, Jan 14 2016
%o (PARI) A080513(n)=n\/2+1 \\ _M. F. Hasler_, Feb 14 2019
%Y Cf. A110654, A266843.
%K nonn,easy
%O 0,2
%A _Robert Price_, Jan 04 2016
%E Simpler definition from _M. F. Hasler_, Feb 14 2019
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