Problem 56593. List the nth term of Rozhenko’s inventory sequence

ChrisR

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Consider a sequence constructed by repeated inventories. A new inventory begins each time a zero is encountered. The first few inventories are
0
1, 1, 0
2, 2, 2, 0
3, 2, 4, 1, 1, 0
4, 4, 4, 1, 4, 0
When the sequence is empty, there are zero 0s. We start a new inventory on the second line—looking at all numbers written so far: one 0, one 1, zero 2s. The zero triggers a new inventory, and the third line reports two 0s, two 1s, two 2s (from the beginning of the third line), and zero 3s. And so on. The sequence then is the rows strung together. For example, the 19th term is 4. 
This sequence produces interesting plots and music. See the related Numberphile video for more. 
Write a function to report the nth term of this sequence. 