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Conway Sequence

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.

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Tag(s) : Mathematics, Fun/Miscellaneous

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# Conway Sequence

## Next Look-and-Say Term Generator

 Calcul Next Term Previous Term

## Conway Sequence Generator

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.

### How does the Conway Sequence works?

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:

1one 111
11two 1s21
21one 2 and one 11211
1211one 1, one 2 and two 1s111221
111221three 1s, two 2s and 1312211

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.

### Can the sequence contain '333'?

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.

### What are the variants of the Conway Sequence?

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.

### Why the sequence is called the Conway Sequence?

This sequence has been invented and analyzed by famous mathematician John H. Conway.

### How to code Conway in Javascript?

// Yves PRATTER// Version 1.0 - 2011/11/07function 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;}

## Source code

dCode retains ownership of the source code of the script Conway Sequence online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Conway Sequence script for offline use on PC, iPhone or Android, ask for price quote on contact page !