Tetration except each repeated exponent increases by 1

[Saracen Rue]

TL;DR Summary

Proper notation to use for $a^{{{{{{(a+1)}^{(a+2)}}^{(a+3)}}^{.}}^{.}}^{.}}$?

I was just wondering what the proper notation would be to describe an infinite power tower which has each repeated exponent increasing by a value of 1, like such;

$$a^{{{{{{(a+1)}^{(a+2)}}^{(a+3)}}^{.}}^{.}}^{.}}$$

 

[fresh_42]

Wikipedia has a couple of notations:
https://en.wikipedia.org/wiki/Tetration
but I wouldn't rely on them in specific cases where an author might have defined his own.