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.