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

0,4

Comments

This sequence is A000012 and A016813 interspersed.

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

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

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)

Example

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)

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}]

Crossrefs

Cf. A000012, A016813, A271343.

Sequence in context: A115521 A140705 A336053 * A376018 A147423 A147085

Adjacent sequences: A266069 A266070 A266071 * A266073 A266074 A266075

Keyword

nonn,easy

Author

Robert Price, Dec 20 2015

Status

approved