A266069 Decimal representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.

1, 4, 2, 121, 4, 2035, 8, 32743, 16, 524239, 32, 8388511, 64, 134217535, 128, 2147483263, 256, 34359737599, 512, 549755812351, 1024, 8796093019135, 2048, 140737488349183, 4096, 2251799813672959, 8192, 36028797018939391, 16384, 576460752303374335, 32768

Offset: 0

Author

Robert Price, Dec 20 2015

Keywords

• nonn
• easy

Comments

Rule 35 also generates this sequence.

Examples

			From _Michael De Vlieger_, Dec 21 2015: (Start)
First 8 rows, replacing leading zeros with ".", the row converted to its binary, then decimal equivalent at right:
              1                =               1 ->     1
            1 0 0              =             100 ->     4
          . . . 1 0            =              10 ->     2
        1 1 1 1 0 0 1          =         1111001 ->   121
      . . . . . . 1 0 0        =             100 ->     4
    1 1 1 1 1 1 1 0 0 1 1      =     11111110011 ->  2035
  . . . . . . . . . 1 0 0 0    =            1000 ->     8
1 1 1 1 1 1 1 1 1 1 0 0 1 1 1  = 111111111100111 -> 32743
(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. A263428, A266068, A266070, A266071, A081253, A266072, A247375, A266073, A266074.

Programs

Mathematica

rule = 3; rows = 30; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]],2], {k,1,rows}]

Python

print([2*4**n - 3*2**((n-1)//2) - 1 if n%2 else 2**(n//2) for n in range(30)]) # Karl V. Keller, Jr., Aug 25 2021

Formula

G.f.: (1+4*x-17*x^2+45*x^3+16*x^4-64*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)). - Colin Barker, Dec 21 2015 and Apr 18 2019

a(n) = 2*4^n - 3*2^((n-1)/2) - 1 for odd n; a(n) = 2^(n/2) for even n. - Karl V. Keller, Jr., Aug 25 2021