A265283 Number of ON (black) cells in the n-th iteration of the "Rule 94" elementary cellular automaton starting with a single ON (black) cell.

1, 3, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 46, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 58, 60, 60, 62, 62, 64, 64, 66, 66, 68

Offset: 0

Author

Robert Price, Dec 06 2015

Keywords

• nonn
• easy

Comments

From Gus Wiseman, Apr 13 2019: (Start)

Also the number of integer partitions of n + 3 such that lesser of the maximum part and the number of parts is 2. The Heinz numbers of these partitions are given by A325229. For example, the a(0) = 1 through a(7) = 10 partitions are:

(21) (22) (32) (33) (43) (44) (54) (55)

(31) (41) (42) (52) (53) (63) (64)

(211) (221) (51) (61) (62) (72) (73)

(2111) (222) (2221) (71) (81) (82)

(2211) (22111) (2222) (22221) (91)

(21111) (211111) (22211) (222111) (22222)

(221111) (2211111) (222211)

(2111111) (21111111) (2221111)

(22111111)

(211111111)

(End)

Examples

			From _Michael De Vlieger_, Dec 14 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                        =  3
                    1 1 . 1 1                      =  4
                  1 1 1 . 1 1 1                    =  6
                1 1 . 1 . 1 . 1 1                  =  6
              1 1 1 . 1 . 1 . 1 1 1                =  8
            1 1 . 1 . 1 . 1 . 1 . 1 1              =  8
          1 1 1 . 1 . 1 . 1 . 1 . 1 1 1            = 10
        1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1          = 10
      1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1        = 12
    1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1      = 12
  1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1    = 14
1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1  = 14
(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
• FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata

Crossrefs

Column k = 2 of A325227.

Cf. A004526, A051924, A118102, A263297, A325193, A325225, A325228, A325229, A325232.

Programs

Mathematica

rule = 94; 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}]
Total /@ CellularAutomaton[94, {{1}, 0}, 65] (* Michael De Vlieger, Dec 14 2015 *)

Formula

Conjectures from Colin Barker, Dec 07 2015 and Apr 16 2019: (Start)

a(n) = (5-(-1)^n+2*n)/2 = A213222(n+3) for n>1.

a(n) = n+2 for n>1 and even.

a(n) = n+3 for n>1 and odd.

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

G.f.: (1+2*x-x^4) / ((1-x)^2*(1+x)).

(End)