User blog:All Things Logical/ABACABADABACABA numbers

ABACABADABACABA Numbers are numbers in the form of 0102010... These numbers follow a rule: the next iteration is the current iteration, with the next number in the middle. Let us construct some base 10 ABACABADABACABA numbers.

Iteration 0: 0

Iteration 1: 010

Iteration 2: 0102​010

Iteration 3: 0102010​3​0102010

etc...

This is the largest possible ABACABADABACABA number in base 10:

010201030102010401020103010201050102010301020104010201030102010601020103010201040102010301020105010201030102010401020103010201070102010301020104010201030102010501020103010201040102010301020106010201030102010401020103010201050102010301020104010201030102010801020103010201040102010301020105010201030102010401020103010201060102010301020104010201030102010501020103010201040102010301020107010201030102010401020103010201050102010301020104010201030102010601020103010201040102010301020105010201030102010401020103010201090102010301020104010201030102010501020103010201040102010301020106010201030102010401020103010201050102010301020104010201030102010701020103010201040102010301020105010201030102010401020103010201060102010301020104010201030102010501020103010201040102010301020108010201030102010401020103010201050102010301020104010201030102010601020103010201040102010301020105010201030102010401020103010201070102010301020104010201030102010501020103010201040102010301020106010201030102010401020103010201050102010301020104010201030102010

The ABACABADABACABA sequence is the sequence of the largest ABACABADABACABA numbers you can make, starting at base 2:

010 in base 2 = 2

0102010 in base 3 = 300

etc...

The sequence would look like this: {2, 300, ...}