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

1, 4, 8, 13, 20, 27, 37, 46, 59, 70, 86, 99, 118, 133, 155, 172, 197, 216, 244, 265, 296, 319, 353, 378, 415, 442, 482, 511, 554, 585, 631, 664, 713, 748, 800, 837, 892, 931, 989, 1030, 1091, 1134, 1198, 1243, 1310, 1357, 1427, 1476, 1549, 1600, 1676, 1729

Offset: 0

Author

Robert Price, Dec 07 2015

Keywords

• nonn
• easy

Examples

			From _Michael De Vlieger_, Dec 09 2015: (Start)
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:
                        1                          =  1 ->   1
                      1 1 1                        =  3 ->   4
                    1 1 1 . 1                      =  4 ->   8
                  1 1 1 . . 1 1                    =  5 ->  13
                1 1 1 . 1 1 1 . 1                  =  7 ->  20
              1 1 1 . . 1 1 . . 1 1                =  7 ->  27
            1 1 1 . 1 1 1 . 1 1 1 . 1              = 10 ->  37
          1 1 1 . . 1 1 . . 1 1 . . 1 1            =  9 ->  46
        1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1          = 13 ->  59
      1 1 1 . . 1 1 . . 1 1 . . 1 1 . . 1 1        = 11 ->  70
    1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1      = 16 ->  86
  1 1 1 . . 1 1 . . 1 1 . . 1 1 . . 1 1 . . 1 1    = 13 ->  99
1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1  = 19 -> 118
(End)
		

References

• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

• Robert Price, Table of n, a(n) for n = 0..999
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A071037.

Programs

Mathematica

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}]
Accumulate[Count[#, n_ /; n == 1] & /@ CellularAutomaton[158, {{1}, 0}, 51]] (* Michael De Vlieger, Dec 09 2015 *)

Formula

Conjectures from Colin Barker, Dec 07 2015 and Apr 18 2019: (Start)

a(n) = 1/16*(10*n^2+2*(-1)^n*n+34*n-3*(-1)^n+19).

a(n) = 1/16*(10*n^2+36*n+16) for n even.

a(n) = 1/16*(10*n^2+32*n+22) for n odd.

a(n) = 2*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4.

G.f.: (1+3*x+2*x^2-x^3) / ((1-x)^3*(1+x)^2).

(End)