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Conway Sequence

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.

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Conway Sequence -

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

### What is the Conway sequence? (Definition)

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.

### 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

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.

### 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.

### Can the sequence contain '4'?

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.

### 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 is the sequence called the Conway Sequence?

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

### 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;}

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Conway Sequence on dCode.fr [online website], retrieved on 2023-09-27, https://www.dcode.fr/conway-sequence

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