A070952 -id:A070952 - cp's OEIS frontend

A261299 Binary representation of the middle column of the "Rule 30" elementary cellular automaton starting with a single ON (black) cell.

1, 11, 110, 1101, 11011, 110111, 1101110, 11011100, 110111001, 1101110011, 11011100110, 110111001100, 1101110011000, 11011100110001, 110111001100010, 1101110011000101, 11011100110001011, 110111001100010110, 1101110011000101100, 11011100110001011001

Offset: 0

Robert Price, Dec 05 2015

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

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

Cf. A070950, A245549, A110240, A051023, A092539, A070952, A110267.

A261299list[nmax_]:=With[{ca=CellularAutomaton[30,{{1},0},{nmax,{{0}}}]},Array[FromDigits[Take[ca,#]]&,nmax+1]];A261299list[25] (* Paolo Xausa, May 30 2023 *)

A265224 Total number of OFF (white) cells after n iterations of the "Rule 30" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 0, 2, 3, 8, 10, 18, 21, 31, 38, 48, 57, 70, 78, 94, 103, 121, 137, 154, 169, 189, 209, 231, 250, 273, 297, 324, 346, 373, 398, 428, 452, 491, 519, 559, 584, 625, 656, 695, 729, 763, 805, 845, 883, 934, 970, 1021, 1065, 1118, 1164, 1222, 1266, 1319, 1366

Offset: 0

Robert Price, Dec 05 2015

Rule 86, the mirror image of Rule 30, also generates this sequence. - Karl V. Keller, Jr., Sep 20 2020

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

Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Eric Weisstein's World of Mathematics, Rule 30
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A051023, A070950, A070951, A070952, A092539, A110240, A110267, A245549.

A309441 Number of ON (black) cells in the n-th iteration of the "honeycomb" elementary cellular automaton with rule 4124, starting with a single ON (black) cell (see Comments for precise definition).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 4, 4, 4, 6, 4, 6, 8, 2, 4, 4, 4, 8, 4, 8, 8, 8, 12, 8, 12, 14, 4, 8, 8, 6, 12, 8, 12, 16, 12, 18, 12, 14, 16, 12, 16, 10, 8, 16, 12, 16, 20, 12, 16, 20, 12, 20, 16, 16, 24, 18, 24, 20, 16, 16, 16, 24, 24, 20, 20, 22, 24, 20, 32, 30, 20, 20

Offset: 0

Rémy Sigrist, Aug 03 2019

nonn

We consider one-dimensional cellular automata:
- where each cell is either ON (black) or OFF (white),
- cells of (n+1)-th generation are offset by half a unit compared to cells of n-th generation, as in a honeycomb:
                           / \ / \ / \ / \
    n-th generation    ...| A | B | C | D |...
                           \ / \ / \ / \ / \
    (n+1)-th generation  ...|   | E |   |   |...
                             \ / \ / \ / \ /
- each cell of (n+1)-th generation is determined by the pattern formed by 4 neighboring cells of n-th generation: the state of cell E is determined by the pattern ABCD,
- if we represent ON cells by 1's and OFF cells by 0's, then we can uniquely represent the set of 16 rules that defines such an automaton by an integer R in the range 0..2^16-1,
- this encoding scheme is similar to that of elementary cellular automata proposed by Stephen Wolfram.
This sequence is based on rule 4124:
- 4124 = 2^12 + 2^4 + 2^3 + 2^2,
- in binary, 12, 4, 3 and 2 are: "1100", "0100", "0011", and "0010",
- these are the patterns (ABCD) that lead to a ON cell in next generation,
- all other patterns lead to an OFF cell.
Starting from a unique ON cell, we will never have 3 consecutive ON cells in subsequent generations.
The ON cells form a binary tree:
- the two lateral branches are infinite,
- are there other infinite branches?

			The first terms, alongside the corresponding generation (with dots instead of 0's and lateral 0's removed for readability), are:
  n  a(n)   n-th generation
  -- ----   ---------------
   0    1                  1
   1    2                 1 1
   2    2                1 . 1
   3    2               1 . . 1
   4    4              1 1 . 1 1
   5    2             1 . . . . 1
   6    4            1 1 . . . 1 1
   7    4           1 . 1 . . 1 . 1
   8    4          1 . . 1 . 1 . . 1
   9    6         1 1 . 1 . . 1 . 1 1
  10    4        1 . . . 1 . 1 . . . 1
  11    6       1 1 . . 1 . . 1 . . 1 1
  12    8      1 . 1 . 1 1 . 1 1 . 1 . 1
  13    2     1 . . . . . . . . . . . . 1
  14    4    1 1 . . . . . . . . . . . 1 1
  15    4   1 . 1 . . . . . . . . . . 1 . 1

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Representation of the iterations for n = 0..999
Rémy Sigrist, PARI program for A309441
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata

Cf. A070952.

PARI
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See Links section.
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a(n) is even for any n > 0 (for symmetry reasons).

a(n) <= n+1 - floor((n+1)/3).

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A261299 Binary representation of the middle column of the "Rule 30" elementary cellular automaton starting with a single ON (black) cell.

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A265224 Total number of OFF (white) cells after n iterations of the "Rule 30" elementary cellular automaton starting with a single ON (black) cell.

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A309441 Number of ON (black) cells in the n-th iteration of the "honeycomb" elementary cellular automaton with rule 4124, starting with a single ON (black) cell (see Comments for precise definition).

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