Infinite summation involving square root of 5

Hello.

I have this recurring function or whatever you call it:

Once increases, we have , inside of which there is another added and so forth (this clearly converges).

As the title says, I am looking forward to obtaining

P.S. I started from the greatest depth and noticed , however later on I am not able to control the expression...

P.P.S. Messing around I've got a good feeling that such infinite summations can be defined like that, no clue how to prove it though