An Interesting Example Of Arithmetic Sequence

Consider a two-letter word AB, where A = APPLE, and B =BOY,
Every time you see A replace it by APPLE and B by BOY, One obtains the following chain:
AB –>APPLEBOY –> APPLEPPLEBOYOY –> APPLEPPLEPPLEBOYOYOY –> …
The length of each string goes like this
2 –> 8 –> 14 –> 20 –>…
It is not difficult to
a_n = a + (n-1) d,
where a = 2, and d=6.