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A265382 Total number of ON (black) cells after n iterations of the "Rule 158" elementary cellular automaton starting with a single ON (black) cell.

%I A265382 #19 Feb 16 2025 08:33:27
%S A265382 1,4,8,13,20,27,37,46,59,70,86,99,118,133,155,172,197,216,244,265,296,
%T A265382 319,353,378,415,442,482,511,554,585,631,664,713,748,800,837,892,931,
%U A265382 989,1030,1091,1134,1198,1243,1310,1357,1427,1476,1549,1600,1676,1729
%N A265382 Total number of ON (black) cells after n iterations of the "Rule 158" elementary cellular automaton starting with a single ON (black) cell.
%D A265382 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A265382 Robert Price, <a href="/A265382/b265382.txt">Table of n, a(n) for n = 0..999</a>
%H A265382 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A265382 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A265382 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A265382 Conjectures from _Colin Barker_, Dec 07 2015 and Apr 18 2019: (Start)
%F A265382 a(n) = 1/16*(10*n^2+2*(-1)^n*n+34*n-3*(-1)^n+19).
%F A265382 a(n) = 1/16*(10*n^2+36*n+16) for n even.
%F A265382 a(n) = 1/16*(10*n^2+32*n+22) for n odd.
%F A265382 a(n) = 2*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4.
%F A265382 G.f.: (1+3*x+2*x^2-x^3) / ((1-x)^3*(1+x)^2).
%F A265382 (End)
%e A265382 From _Michael De Vlieger_, Dec 09 2015: (Start)
%e A265382 First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row, and the running total up to that row:
%e A265382                         1                          =  1 ->   1
%e A265382                       1 1 1                        =  3 ->   4
%e A265382                     1 1 1 . 1                      =  4 ->   8
%e A265382                   1 1 1 . . 1 1                    =  5 ->  13
%e A265382                 1 1 1 . 1 1 1 . 1                  =  7 ->  20
%e A265382               1 1 1 . . 1 1 . . 1 1                =  7 ->  27
%e A265382             1 1 1 . 1 1 1 . 1 1 1 . 1              = 10 ->  37
%e A265382           1 1 1 . . 1 1 . . 1 1 . . 1 1            =  9 ->  46
%e A265382         1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1          = 13 ->  59
%e A265382       1 1 1 . . 1 1 . . 1 1 . . 1 1 . . 1 1        = 11 ->  70
%e A265382     1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1      = 16 ->  86
%e A265382   1 1 1 . . 1 1 . . 1 1 . . 1 1 . . 1 1 . . 1 1    = 13 ->  99
%e A265382 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1  = 19 -> 118
%e A265382 (End)
%t A265382 rule = 158; rows = 30; Table[Total[Take[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}],k]],{k,1,rows}]
%t A265382 Accumulate[Count[#, n_ /; n == 1] & /@ CellularAutomaton[158, {{1}, 0}, 51]] (* _Michael De Vlieger_, Dec 09 2015 *)
%Y A265382 Cf. A071037.
%K A265382 nonn,easy
%O A265382 0,2
%A A265382 _Robert Price_, Dec 07 2015