I have a sequence of numbers here, which follow a very simple rule that can be expressed using a single fraction:
1, 2, 10, 11, 13, 15, 20, 22, 25, 28, ...
What is this rule (in particular, what is the fraction), and what are the next few numbers in the sequence?
Since nobody's gotten it yet, I'll add a few hints over the next little while.
(06-16 13:22) 1. The numbers in the sequence never get more than 2 digits long.
The items in the sequence should be numbered from 2 onwards, and then item $b$ is
the base-$b$ representation of $\displaystyle \left \lfloor \frac{b^2}{4} \right \rfloor$, which can also be written as just $\displaystyle \left \lfloor \frac{100}{4} \right \rfloor$, since $b$ in base $b$ is always $10$.
So the fraction that defines the sequence is
100/4
and the next few entries in the sequence are:
30, 33, 37, 3B, 40, 44, 49, 4E, 50.
1, 2, 10, 11, 13, 15, 20, 22, 25, 28, ...
1, 2, 10, 11, 13, 15, 20, 22, 25, 28, 31, 100, 103, 107, 111, 115, 119, 110, ...
The pattern is explained as follows:
Add (1) to first number (1) time, then add a zero to first number
1, 2, 10
Add (1) to new number once
11
Add (1+1) to new number (1+1) times, then add a zero to second number
13, 15, 20
Add (1+1) to new number once
22
Add (1+1+1) to new number (1+1+1) times, then add a zero to third number
25, 28, 31, 100
Add (1+1+1) to new number once
103
Add (1+1+1+1) to new number (1+1+1+1) times, then add a zero to forth number
107, 111, 115, 119, 110
