Instead of thinking about frame duplication, try drawing a
timing chart, so you know where the button should be in each frame.
If you must do it via duplication: duplicate more frames towards the extremes, since that's where it's slowest, e.g. no dupes of the middle two frames, 1 dupe each of the frames next to the middle frames, 2 dupes each of the next 2. That will give you sine-like motion. However, that won't give you the overshooting that'll make it feel springy, that's tough to do well without doing a timing chart.