A266072 Number of ON (black) cells in the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.

1, 1, 1, 5, 1, 9, 1, 13, 1, 17, 1, 21, 1, 25, 1, 29, 1, 33, 1, 37, 1, 41, 1, 45, 1, 49, 1, 53, 1, 57, 1, 61, 1, 65, 1, 69, 1, 73, 1, 77, 1, 81, 1, 85, 1, 89, 1, 93, 1, 97, 1, 101, 1, 105, 1, 109, 1, 113, 1, 117, 1, 121, 1, 125, 1, 129, 1, 133, 1, 137, 1, 141

Offset: 0

Author

Robert Price, Dec 20 2015

Keywords

• nonn
• easy

Comments

This sequence is A000012 and A016813 interspersed.

Also column 1 of A271343. - Omar E. Pol, Apr 06 2016

Examples

			From _Michael De Vlieger_, Dec 21 2015: (Start)
First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row:
                      1                        =  1
                    1 . .                      =  1
                  . . . 1 .                    =  1
                1 1 1 1 . . 1                  =  5
              . . . . . . 1 . .                =  1
            1 1 1 1 1 1 1 . . 1 1              =  9
          . . . . . . . . . 1 . . .            =  1
        1 1 1 1 1 1 1 1 1 1 . . 1 1 1          = 13
      . . . . . . . . . . . . 1 . . . .        =  1
    1 1 1 1 1 1 1 1 1 1 1 1 1 . . 1 1 1 1      = 17
  . . . . . . . . . . . . . . . 1 . . . . .    =  1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . 1 1 1 1 1  = 21
(End)
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2000
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A000012, A016813, A271343.

Programs

Mathematica

rule = 3; 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}]

Formula

Conjectured g.f.: (1 + x - x^2 + 3*x^3)/(-1 + x^2)^2. - Michael De Vlieger, Dec 21 2015

Conjectures from Colin Barker, Dec 21 2015: (Start)

a(n) = n-(-1)^n*(n-1).

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