How Many Widgets in Santa's Bag?

Widgets have sizes A through E (A < B < C < D < E), and there's a huge demand only for A-size widgets. But at present, Santa's only widget is E-size!

The elf in charge of widget production, being both bored and somewhat maniacal, proposes the following scheme to put some A-size widgets into Santa's bag:

At time t = 0, only the E-size widget is in the bag. At each time t = 1,2,3,... (until all widgets in the bag are A-size), a smallest possible widget of size larger than A is discarded from the bag and is replaced with t+1 widgets of the next-smaller size.

Here's a picture:

t bag -

My! That does grow quickly... If my string size didn't let me down, I get 33164 'A' widgets, at time 397.

Partial solution:

t A B C D E

0 0 0 0 0 1 1 0 0 0 2 0 2 0 0 3 1 0 3 0 4 2 1 0 ... 7 26 0 2 1 0 8 26 9 1 1 0 ... 17 152 0 1 1 0 18 152 19 0 1 0 ... 37 703 0 0 1 0 38 703 0 39 0 0

I agree - that makes me feel better about my construction!

Now, there must be a closed-form solution, probably with factorials...

'Phil Carmody' wrote ...

Exactly!

To me, the interesting thing is the rate of growth of these numbers with increasing size of the widget initially in the bag

look at you line 79 and 80. 80 is wrong...unless i am confused...which is possible tonight !!

162*2^38 -2 =44530220924926. I hope I got it now.

Well nice one.

I quickly did it.

I probably missed a few of my calculations since I was hasty, but I found that time = 247*2^26 -2

Ill do it again a bit more slowly to make sure....and to re-read the question more profoundly.