DES key expansion: why are these two descriptions different?
I'm not going to look up the DES key schedule such, but the connection between your two sequences

$$( a_i ) = ( 1, 2, 4, 6, 8, 10, 12, 14, 15, 17, 19, 21, 23, 25, 27, 28 )$$

and

$$( b_i ) = ( 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1 )$$

is obvious: $b_i = a_i - a_{i - 1}$ (with $a_0$ taken to be 0).

That is, the second sequence gives the difference between successive elements of the first sequence.



Edit: OK, I did look it up, see link above.  Basically, the first sequence gives the number of bits by which the halves of the original key need to be rotated to obtain the $i$-th subkey, whereas the second sequence gives the number of bits by which the halves of the previous subkey need to be rotated to obtain the next one.  That's all there is to it.