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

Conway Sequence - dCode

Tag(s) : Mathematics, Fun/Miscellaneous

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

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 |

__Example:__ The **Conway sequence** is 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... (and is often used as a riddle where the player must guess the next term)

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

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