A267708 Triangle read by rows giving successive states of cellular automaton generated by "Rule 206" initiated with a single ON (black) cell.

1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1

Offset

0

Comments

Row n has length 2n+1.

This sequence is also generated by Rule 238.

Links

Robert Price, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

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

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Formula

Conjecture: a(n) = (1 + (-1)^floor(2*sqrt(n)))/2. - Velin Yanev, Nov 30 2016

a(n) = A351319(n) for n != 2. - Chai Wah Wu, Jul 29 2022

Example

The first ten rows:
                  1
                1 1 0
              1 1 1 0 0
            1 1 1 1 0 0 0
          1 1 1 1 1 0 0 0 0
        1 1 1 1 1 1 0 0 0 0 0
      1 1 1 1 1 1 1 0 0 0 0 0 0
    1 1 1 1 1 1 1 1 0 0 0 0 0 0 0
  1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0

Mathematica

rule=206; rows=20; ca=CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]], {rows-k+1, rows+k-1}], {k, 1, rows}]; (* Truncated list of each row *) Flatten[catri] (* Triangle Representation of CA *)

Prog

(Python)
from math import isqrt
def A267708(n): return int((k:=isqrt(n))**2+k-n+1>0) # Chai Wah Wu, Jul 28 2022

Crossrefs

Cf. A109241, A171476, A351319.

Cf. A028242 (run lengths).

Sequence in context: A197774 A219009 A353519 * A167021 A267687 A304653

Adjacent sequences: A267705 A267706 A267707 * A267709 A267710 A267711

Keyword

nonn,tabf,easy

Author

Robert Price, Jan 19 2016

Status

approved