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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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.

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.

`// 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 2022-12-09,

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

https://www.dcode.fr/conway-sequence

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