[seqfan] Three connected sequences

[Ali Sada]

Hi everyone,

I need help defining the three sequences below, if you see them fit for the OEIS.

We have sequenced “slots” and each one contains n balls. We want to empty the slots one by one. 

The ball in the first slot moves 1 step to the second slot. Now, the sum of balls in the second slot is 3. So, a(1) = 3. 

The 3 balls in the second slot move 3 steps to slot 5 raising the number there to 8. So, a(2) = 8.

The third slot has 3 balls and they move 3 steps to slot 6. So, a(3) = 9.

The fourth slot has 4 balls and they move 4 steps to slot 8. So, a(4) = 12

The fifth slot has 8 balls (including the ones from slots 1 and 2). They move 8 steps and merge with the balls in slot 13. So, a(5) = 21. 

And so on. 

This is the sequence we get

3, 8, 9, 12, 21, 24, 21, 32, 27, 42, 33, 36, 55, 56, 63, 48, 72, 72, 57, 104, 63, 88, 69, 96, 75, 78, 81, 84, 87, 126, 135, 128, 99, 178, 147, 108, 111, 152, 165, 168, 123, 168, 129, 132, 189, 184, 141, 144, 147, 200, 216, 208, 231, 216, 231, 224, 171, 232, 177, 312
  
Now, if the initial balls in each slot had a distinctive color (the ball in slot one was green, the two balls in slot 2 were blue, etc.), what is the highest number of colors we could get in each slot?

This is the sequence we get 
1, 2, 1, 1, 3, 2, 1, 2, 1, 1, 1, 1, 4, 2, 3, 1, 1, 2, 1, 3 

The final sequence is the first appearances in the second sequence. 

I appreciate your responses, as usual.
 
Best,

Ali