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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) invented by mathematician John H. Conway. 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

 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) invented by mathematician John H. Conway. 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, it must use the previous one, read it digit by digit and locate 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, ...

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'?

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 consider a different seed.

For a seed g of 2,3,4,5,6,7,8,9 or 0, these values are obtained:

Example: 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.

### How to code Conway in Javascript?

// Yves PRATTER// Version 1.0 - 2011/11/07function previousConway(t) { r = ""; // impossible if (t.length%2 == 1) return r; 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;}