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A265722 Number of ON (black) cells in the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

%I A265722 #21 Feb 16 2025 08:33:28
%S A265722 1,0,1,4,1,8,1,12,1,16,1,20,1,24,1,28,1,32,1,36,1,40,1,44,1,48,1,52,1,
%T A265722 56,1,60,1,64,1,68,1,72,1,76,1,80,1,84,1,88,1,92,1,96,1,100,1,104,1,
%U A265722 108,1,112,1,116,1,120,1,124,1,128,1,132,1,136,1,140
%N A265722 Number of ON (black) cells in the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.
%D A265722 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A265722 Colin Barker, <a href="/A265722/b265722.txt">Table of n, a(n) for n = 0..1000</a>
%H A265722 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A265722 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A265722 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A265722 Conjectures from _Colin Barker_, Dec 14 2015 and Apr 16 2019: (Start)
%F A265722 a(n) = 1/2*(-2*(-1)^n*n+2*n+3*(-1)^n-1).
%F A265722 a(n) = 2*a(n-2) - a(n-4) for n>3.
%F A265722 G.f.: (1-x^2+4*x^3) / ((1-x)^2*(1+x)^2).
%F A265722 (End)
%F A265722 a(n) = A019425(n), n>1. - _R. J. Mathar_, Jan 10 2016
%e A265722 From _Michael De Vlieger_, Dec 14 2015: (Start)
%e A265722 First 12 rows, replacing zeros with "." for better visibility of ON cells, followed by the total number of 1's per row at right:
%e A265722                       1                        =   1
%e A265722                     . . .                      =   0
%e A265722                   . . 1 . .                    =   1
%e A265722                 1 1 . . . 1 1                  =   4
%e A265722               . . . . 1 . . . .                =   1
%e A265722             1 1 1 1 . . . 1 1 1 1              =   8
%e A265722           . . . . . . 1 . . . . . .            =   1
%e A265722         1 1 1 1 1 1 . . . 1 1 1 1 1 1          =  12
%e A265722       . . . . . . . . 1 . . . . . . . .        =   1
%e A265722     1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1      =  16
%e A265722   . . . . . . . . . . 1 . . . . . . . . . .    =   1
%e A265722 1 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 1  =  20
%e A265722 (End)
%t A265722 rule = 1; rows = 30; Table[Total[Table[Take[CellularAutomaton[rule, {{1},0},rows-1,{All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]
%Y A265722 Cf. A265718, A265720, A265721.
%K A265722 nonn,easy
%O A265722 0,4
%A A265722 _Robert Price_, Dec 14 2015