[seqfan] Re: Three connected sequences

[Allan Wechsler]

Hi, Ali; it's good to hear from you again.

When you invent a process like this, it's best to concentrate on the
*simplest* sequences that you can read off the sequence.

In your process, you start with B(n) = n for all n. Then, at step i, you
look at B(i), and you increment B(i + B(i)) by B(i).

Your first sequence is "How many balls are in the target bucket just after
step i?" But it's simpler to ask "How many balls do I move at step i?"

I get 1,3,3,4,8,9,7,12..., and because I picked the simplest imaginable
sequence related to this process it turned out that somebody had thought of
the process before. That was Ctibor O. Zizka, and he created
https://oeis.org/A137417 in 2008. There are interesting comments there, and
the behavior of the process raises many questions! Be sure to look at the
extremely thoughtprovoking scatterplot of the first two million terms --
it's mindboggling.

If you decide to introduce your interesting new sequences, make sure to
link them to A137417. Your "highest number of colors" question is an
interesting twist on a question that is already asked there, "how many
times is each bucket incremented?".

On Tue, Jul 20, 2021 at 4:58 AM Ali Sada via SeqFan <seqfan at list.seqfan.eu>
wrote:

> 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
>
>
>
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