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

0,2

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

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)

Example

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)

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 *)

Crossrefs

Cf. A071037.

Sequence in context: A312218 A350304 A368612 * A056738 A170907 A143978

Adjacent sequences: A265379 A265380 A265381 * A265383 A265384 A265385

Keyword

nonn,easy

Author

Robert Price, Dec 07 2015

Status

approved