cp's OEIS Frontend

To generate the value for n, write out n's decimal expansion. Then, write out 1's decimal expansion (0000000000....001). Compute how many times you need to change 0 to a 1 or a 1 to a 0 in order to switch from one number to the other.

The value for 2^x is always 2. The value for 2^x +1 is always 1. The value for 2^x -1 is always x-1 when x > 0. To get to 2^x, you need to drop the 1 at the beginning and add the 1 in the 2^x place value.

For 2^x + 1, you need to add the 1 in the 2^n place value, but you keep the 1 in the 1s place value. Thus you are only adding or getting rid of 1 digit.

For 2^x -1, it will have x digits, and all of them will be 1's. You already have 1 in the 1's place value, so there are n-1 digits left over.

A four-way fair share sequence. This is similar to the Thue-Morse Sequence. The Thue-Morse Sequence is the fairest way to split objects amongst two groups. If we call the groups A and B, most people split ABABABABABABABABABABABAB.......

This is unfair for B, because out of the best 2, A gets the best. Out of the second best 2, a gets the best. The Thue-Morse Sequence solves this:

ABBABAABBAABABBABAABABBAABBABAAB... The easiest way to generate the Thue-Morse Sequence is starting with a 1. Every 1 becomes 12. Every 2 becomes 21. Thus the sequence is obtained by recursion.

The present sequence is the same, but for splitting objects amongst 4 groups. Start with a 1. Every 1 becomes 1,2,3,4. Every 2 becomes 2,3,4,1. Every 3 becomes 3,4,1,2. Every 4 becomes 4,1,2,3.