Read up on the OFFSET() function to fully understand it.
=OFFSET(StartingRange, OffsetRow(s), OffsetColumn(s), NumberOfRows, NumberofColumns)
The first version used the 2nd and 4th parameters to do "row" offsets and sizings. You would use the 3rd and 5th parameters instead:
Enter this starting formula in a cell and copy to the right:
=SUM(OFFSET($A$1, ,(COLUMN(A1)*4)-4, , 4))