A363343 Array read by ascending antidiagonals: T(n,k) is the value of the k-th cell on the n-th diagonal from the right of rule-30 1-D cellular automaton, when started from a single ON cell, with n, k >= 1.

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

Offset

1

Comments

Diagonals from the right are periodic, with periods (A094605) being a power of 2 and doubling at apparently non-predictable intervals.

Links

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened).

Michael Brunnbauer, Diagonals in elementary cellular automaton 30, 2019 (local PDF copy, with author's permission).

Eric S. Rowland, Local Nested Structure in Rule 30, Complex Systems 16 (2006), pp. 239-258.

Eric Weisstein's World of Mathematics, Rule 30.

Index entries for sequences related to cellular automata

Example

The following diagram illustrates how the array is built.
.
            1
             \
          1 1 1
           \ \ \
        1 1 0 0 1
         \ \ \ \ \
      1 1 0 1 1 1 1
         \ \ \ \ \ \
    1 1 0 0 1 0 0 0 1
           \ \ \ \ \ \	
  1 1 0 1 1 1 1 0 1 1 1     Array begins:
             \ \ \ \ \ \___ 1 1 1 1 1 1 1 1 1 1 1 1 ... (period 1)
       ...    \ \ \ \ \____ 1 0 1 0 1 0 1 0 1 0 1 0 ... (period 2)
               \ \ \ \_____ 1 0 1 0 1 0 1 0 1 0 1 0 ... (period 2)
                \ \ \______ 1 1 0 0 1 1 0 0 1 1 0 0 ... (period 4)
                 \ \_______ 1 0 1 1 0 1 0 0 1 0 1 1 ... (period 8)
                  \________ 1 0 1 0 1 0 0 0 1 0 1 0 ... (period 8)
                            ...

Mathematica

A363343list[dmax_]:=Module[{ca=CellularAutomaton[30, {{1}, 0}, dmax-1], a}, a=Array[Drop[Diagonal[ca, #], Floor[(dmax-#)/2]]&, dmax, 0]; Array[Diagonal[a, #]&, dmax, 1-dmax]]; A363343list[15] (* Generates 15 antidiagonals *)

Crossrefs

Cf. A070950.

Cf. A094605 (periods of diagonals), A363344 (diagonals from the left).

Sequence in context: A070909 A115954 A115526 * A336923 A239681 A054524

Adjacent sequences: A363340 A363341 A363342 * A363344 A363345 A363346

Keyword

nonn,tabl

Author

Paolo Xausa, May 28 2023

Status

approved