A265721 Decimal representation of the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

1, 0, 4, 99, 16, 1935, 64, 32319, 256, 522495, 1024, 8381439, 4096, 134189055, 16384, 2147368959, 65536, 34359279615, 262144, 549753978879, 1048576, 8796085682175, 4194304, 140737458995199, 16777216, 2251799696244735, 67108864, 36028796549201919, 268435456

Offset

0,3

Comments

Rule 33 also generates this sequence.

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

Index entries for linear recurrences with constant coefficients, signature (0,21,0,-84,0,64).

Formula

From Colin Barker, Dec 14 2015 and Apr 16 2019: (Start)

a(n) = 21*a(n-2) - 84*a(n-4) + 64*a(n-6) for n>5.

G.f.: (1-17*x^2+99*x^3+16*x^4-144*x^5) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-4*x)*(1+4*x)).

(End)

a(n) = 2*4^n - 7*2^(n-1) - 1 for odd n; a(n) = 2^n for even n. - Karl V. Keller, Jr., Aug 24 2021

Example

From Michael De Vlieger, Dec 14 2015: (Start)
First 8 rows, replacing leading zeros with ".", the row converted to its binary (A265720), then decimal equivalent at right:
              1                ->               1  =     1
            . . 0              ->               0  =     0
          . . 1 0 0            ->             100  =     4
        1 1 0 0 0 1 1          ->         1100011  =    99
      . . . . 1 0 0 0 0        ->           10000  =    16
    1 1 1 1 0 0 0 1 1 1 1      ->     11110001111  =  1935
  . . . . . . 1 0 0 0 0 0 0    ->         1000000  =    64
1 1 1 1 1 1 0 0 0 1 1 1 1 1 1  -> 111111000111111  = 32319
(End)

Mathematica

rule = 1; 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}]

Prog

(Python) print([2*4**n  - 7*2**(n-1) - 1 if n%2 else 2**n for n in range(50)]) # Karl V. Keller, Jr., Aug 24 2021

Crossrefs

Cf. A265718, A265720, A059841, A056830, A000975, A265722, A128918, A265723, A265724.

Sequence in context: A320358 A180830 A361883 * A224475 A352020 A091268

Adjacent sequences: A265718 A265719 A265720 * A265722 A265723 A265724

Keyword

nonn,easy

Author

Robert Price, Dec 14 2015

Status

approved