The Binary Square Diagonal Sequence
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zȯḋm←CtNCİ□ṁḋN


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Continually prints the results as an infinite list.

Explanation

I wonder whether there's a better way to get every nth element from a list than splitting the list into chunks of length n and retrieving the head of each chunk.

      tN          Get a list of all natural numbers except 1. (A)

             N    Get a list of all natural numbers.
           ṁḋ     Convert each to its binary representation and join them 
                  all into a single list.
         İ□       Get a list of squares of all natural numbers.
        C         Cut the list of bits into chunks of corresponding sizes. (B)

zȯ                Zip (A) and (B) together with the following function.
     C            Split the bit list (from B) into chunks of the given length
                  (from A).
   m←             Get the head of each chunk. This is the diagonal of the
                  bit list arranged as a square.
  ḋ               Interpret the resulting bits as binary digits and return
                  the result.


To be clear, we extract the diagonal of an n x n square by splitting its linear form into chunks of length n + 1 and retrieving the first element of each chunk:

[[1 , 0 , 1 , 0
  0],[1 , 0 , 1
  1 , 0],[1 , 0
  0 , 1 , 0],[1]]