Convert the following sequences into logical equivalence series

Below is a sequence I would like to convert into its mathematical representation: 

[1 01 10 010 11 011 101 0101 110 0110 1010 01010 111 0111 1011 01011 1101 01101 10101 010101 1110 01110 10110 010110 11010 011010 101010 0101010 1111 01111 10111 010111 11011 011011 101011 0101011 11101 011101 101101 0101101 110101 0110101 1010101 01010101 11110 011110 101110 0101110 110110 0110110 1010110 01010110 111010 0111010 1011010 01011010 1101010 01101010 10101010 010101010 11111 011111 101111 0101111 110111 0110111 1010111 01010111 111011 0111011 1011011 01011011 1101011 01101011 10101011 010101011 111101 0111101 1011101 01011101 1101101 01101101 10101101 010101101 1110101 01110101 10110101 010110101 11010101 011010101 101010101 0101010101 111110 0111110 1011110 01011110 1101110 01101110 10101110 010101110 .... ]

I would like to develop a "formula" that generates the above series

  • What do you mean by logical equivalence series?

  • What's wrong with the one I gave you?

  • @Rage: I meant rules and formula that would generated the series @Alessandro Iraci: Nothing wrong :) , I reposted it because initially I thought it was incomplete since I reversed binary incorrectly, feel free to repost answer there, too! It was a mistake on my part. I would love to see more about how you formulated the solution, and how you incorporated inverse correlations.

  • Honestly, I pretty much guessed! I noticed that there never were two consecutive 0s, and then the even terms were just sequences of 1s and 10s ordered by length and then antilexicographically. The rest was just a simple computation.

  • No need to take this one too, you can close it, or leave it so maybe someone else will give a simpler formula. :)