Enumerate integers by Hamming weight, modulo bit shifting

Question

I need to sample integers from an ordered array described as follows.

Let k be a positive integer.

• All entries are nonnegative integers in [0,2^k)

• The list starts at 0

• All the (increasing) integers with hamming weight 1 modulo bit shifting (i.e. multiplication by 2) follow.
• All the (increasing) integers with hamming weight 2 modulo bit shifting, follow, and so on.

The array for k=5 looks like this:

        0 ( weight 0 )
        1 ( weight 1 )
       11 ( weight 2 )
      101
     1001
    10001
      111 ( weight 3 )
     1011
     1101
    10011
    10101
    11001 

In particular, given an entry on the list, I would like to deduce the next entry algorithmically.

I know this can be done in several ways (see e.g. this question). For the sake completeness, this is how these other arrays looks like, in contrast to the one above:

        0 ( weight 0 )
        1 ( weight 1 )
       10
      100
     1000
    10000 
       11 ( weight 2 )
      101
      110
  ... etc

Answer 1

I figured out the answer to this, in case anyone needs it.

Given an entry, add the second least bit and carry. If it reduces the Hamming weight, put the necessary number of 1's starting from the second least-significant position.