From description:
The $n^\text{th}$ line of output is the relative lengths of the marks on a ruler subdivided in intervals of $({}^1/_2)^{n}$ of an inch. For example, the fourth line of the output gives the relative lengths of the marks that indicate intervals of one-sixteenth of an inch on a ruler."
So for the first line, the program prints "1" and for the second line it prints "1 2 1".
This is where I am lost, how does "1 2 1" represent subdivision of fourths when there are only 3 numbers. Why is it that to print the next line the recursive relation is $f(n) = f(n - 1) + n + f(n - 1)$ where $f(n)$ prints a string of numbers, and $f(1) = 1$?
I think the text is describing a ruler like this (from Clipart Kid):
The 1/16 mark, not labeled but pictured to the left of the 1/8 mark, is of length 1. Then the lengths of all the marks (ignoring the 0 and 1) are $$1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1$$ (Admittedly the 1/2 mark is not really 8 times as long as the 1/16 mark, but you get the idea.)
This is your $f(n)$. In my opinion, the text is not very clear in the description.