A266982 Triangle read by rows giving successive states of cellular automaton generated by "Rule 81" initiated with a single ON (black) cell.

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

Offset

0

Comments

Row n has length 2n+1.

This sequence is also generated by rule 113.

References

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

Links

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

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Formula

Conjecture: a(0)=1; for n>0, a(n) = ( Sum_{i=1..n-2} floor((n-2)/i) ) mod 2. - Wesley Ivan Hurt, Mar 22 2016

Example

The first ten rows:
                  1
                0 0 1
              1 1 0 0 0
            0 0 1 1 1 1 1
          1 1 0 0 0 0 0 0 0
        0 0 1 1 1 1 1 1 1 1 1
      1 1 0 0 0 0 0 0 0 0 0 0 0
    0 0 1 1 1 1 1 1 1 1 1 1 1 1 1
  1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
The full writeup, including leading and trailing 0's and 1's, gives the following pattern:
00000000000000100000000000000
11111111111110011111111111111
00000000000011000000000000000
11111111111001111111111111111
00000000001100000000000000000
11111111100111111111111111111
00000000110000000000000000000
11111110011111111111111111111
00000011000000000000000000000
11111001111111111111111111111
00001100000000000000000000000
- R. J. Mathar, Aug 07 2025

Mathematica

rule=81; 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 *)

Crossrefs

Sequence in context: A188398 A288929 A285083 * A364746 A051341 A057211

Adjacent sequences: A266979 A266980 A266981 * A266983 A266984 A266985

Keyword

nonn,tabf,easy

Author

Robert Price, Jan 07 2016

Status

approved