Conway Sequence

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.

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).

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.

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.

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.

The Conway sequence is set to begin with 1 by default, but it is possible to use a different seed.

It is possible to use slightly different rules:

— Read the previous term and count all occurrences of numbers, listed in ascending order.

— Read the previous term and count all occurrences of numbers, listed in descending order.

— Read the previous term and count all occurrences of numbers, listed in order of appearance.

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

// 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;
}