So, I have this really weird series. I can easily predict what terms will come next, but I'm having trouble creating a nice formula that will describe this. Here is what I have thus far:

Time 1: (1/6)

Time 2: (1/6)(1/4)

Time 3: (1/6)(1/4)^2 + (1/6)^3

Time 4: (1/6)(1/4)^3 + 2[(1/6)^3 * (1/4)]

Time 5: (1/6)(1/4)^4 + 3[(1/6)^3 * (1/4)^2] + (1/6)^5

Time 6: (1/6)(1/4)^5 + 4[(1/6)^3 * (1/4)^3] + 3[(1/6)^5 * (1/4)]

Time 7: (1/6)(1/4)^6 + 5[(1/6)^3 * (1/4)^4] + 6[(1/6)^5 * (1/4)^2] + (1/6)^7

Time 8: (1/6)(1/4)^7 + 6[(1/6)^3 * (1/4)^5] + 10[(1/6)^5 * (1/4)^3] + 4[(1/6)^7 * (1/4)]

Time 9: (1/6)(1/4)^8 + 7[(1/6)^3 * (1/4)^6] + 15[(1/6)^5 * (1/4)^4] + 10[(1/6)^7 * (1/4)^2] + (1/6)^9

Time 10: (1/6)(1/4)^9 + 8[(1/6)^3 * (1/4)^7] + 21[(1/6)^5 * (1/4)^5] + 20[(1/6)^7 * (1/4)^3] + 5[(1/6)^9 * (1/4)]

...and so on, as time goes to infinity. Now, to each of these, multiply them by 3^n. So, for time 1, it is (1/6)(3^1); for time 2, it is (1/6)(1/4)(3^2), for time 3, it is [(1/6)(1/4)^2 + (1/6)^3](3^3), and so on.

I know that this series is converging to 0.4

However, I want to make this series into a nice formula, so that I can easily evaluate the limit as n goes to infinity. Any hints, tips, or suggestions would be greatly appreciated! (and, if I have messed up anywhere so far, please let me know!!)