I was thinking about the marshmallow patience game where you're given a marshmallow and you can either eat it or wait for several minutes and have two marshmallows instead. Specifically, I was thinking of a generalized n-round variant, where, in any round, you can eat as many marshmallows as you want, and the next round you start with twice what you didn't eat. I then decided on a simple strategy: every round, eat the floor of 1/4 of the marshmallows you have. I then did a bit of math and got three sequences: the number of marshmallows you start each round with, the number you eat each round, and the number you end each round with. I calculated a few values in my head:

```
1 0 1
2 0 2
4 1 3
6 1 5
10 2 8
16 4 12
24 6 18
36 9 27
54 13 41
82 20 62
124 31 93
186 46 140
. . .
. . .
. . .
```

And so I looked all three sequences up in the OEIS. The first two don't seem to be on it, unless they are but start on a different value. However, the third sequence, `1, 2, 3, 5, 8, 12, 18, 27, 41, ...`

, is on it, twice: A061419 and A156623. The former, A061419, had a recursive formula that at first glance didn't seem the same as mine (`a(n) = ceiling(a(n-1)*3/2)`

vs. `a(n) = ceiling(a(n-1)*2-(a(n-1)*2)/4)`

), but I managed to algebra mine into the OEIS one, so yes, they are the exact same sequence.

This was just me messing around with a game and getting a bit of math out of it.