Tool to generate Conway sequences, a sequence of digits (also called Look-and-Say) where each term is made of the reading of the digits of the previous term.

Conway Sequence - dCode

Tag(s) : Mathematics, Fun/Miscellaneous

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The Conway sequence, also known as the look-and-say sequence, is a sequence of integers that is constructed by describing the digits or groups of digits present in the previous sequence. It was discovered by British mathematician John Horton Conway.

To generate the next term in the sequence, use the previous one, by reading it digit by digit and grouping the numbers that are repeated consecutively. The sequence usually begins with `1` first term (also called `seed`).

__Example:__

Term | Is read | Is written |
---|---|---|

1 | one 1 | 11 |

11 | two 1s | 21 |

21 | one 2 and one 1 | 1211 |

1211 | one 1, one 2 and two 1s | 111221 |

111221 | three 1s, two 2s and 1 | 312211 |

The Conway sequence is `1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, …` (and is often used as a riddle, a logic sequence, where the player must guess the next term)

Conway's sequence is also known as the audioactive suite or *look and say* sequence.

The sequence with seed `1` contains only the digits `1`, `2` and `3`.

All terms begin with `1` or `3` except the 3rd.

Reductio ad absurdum (assuming the seed does not contain `333`):

Suppose that `333` appears for the first time at term n, then the term n-1 must also contain `333` (`_333` or `333_` can only appear with a series of three `3` in the previous term). Contradiction, the hypothesis is false, so `333` never appears.

If the term n contained a digit greater than 3, it must come from more than 3 (i.e. 4 or more) equal consecutive digits in the term n-1 which never happens.

It should be noted that by using a seed containing 4 or containing a repetition of 4 digits, then it is possible to generate Conway sequences with additional digits, including the digit `4`.

The Conway sequence is set to begin with `1` by default, but it is possible to use a different seed.

__Example:__ For a seed `g` of 2,3,4,5,6,7,8,9 or 0, the sequence obtained is `g, 1g, 111g, 311g, 13211g, 111312211g …` (the seed is always at the end).

It is possible to use slightly different rules:

— Read the previous term and count all occurrences of numbers, listed in ascending order.

__Example:__ `1, 11, 21, 1112, 3112, 211213, 312213, 212223, 114213, 31121314, 41122314, …`

— Read the previous term and count all occurrences of numbers, listed in descending order.

__Example:__ `1, 11, 21, 1211, 1231, 131221, 132231, 232221, 134211, 14131231, 14231241, …`

— Read the previous term and count all occurrences of numbers, listed in order of appearance.

__Example:__ `1, 11, 21, 1211, 3112, 132112, 311322, 232122, 421311, 14123113 …`

The Conway sequence is similar to run-length encoding.

This sequence has been invented and analyzed by famous mathematician John H. Conway also known for developing The Game of Life.

`// Yves PRATTER`

// Version 1.0 - 2011/11/07

function previousConway(t) {

r = "";

if (t.length%2 == 1) return r;// impossible

idx = 0;

while (idx < t.length){

for(i=0; i < t.charAt(idx); i++) { r += t.charAt(idx+1); }

idx += 2;

}

return r;

}

function conway(t) {

if (t == "") return "0";

r = "";

idx = 0;

while (idx < t.length){

for(i=1; t.charAt(idx+i) == t.charAt(idx); i++) {}

r += i + t.charAt(idx);

idx += i;

}

return r;

}

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Cite as source (bibliography):

*Conway Sequence* on dCode.fr [online website], retrieved on 2023-09-27,

- Next Look-and-Say Term Generator
- Conway Sequence Generator
- What is the Conway sequence? (Definition)
- How does the Conway Sequence works?
- Can the sequence contain '333'?
- Can the sequence contain '4'?
- What are the variants of the Conway Sequence?
- Why is the sequence called the Conway Sequence?
- How to code Conway in Javascript?

sequence,conway,look,say,audioactive,11,21,1211,111221,312211,13112221,1113213211

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