The inventory sequence is an integer sequence generated by repeatedly counting the occurrences of 0, then 1, then 2, and so on among the terms already written. Append each count immediately. Whenever the appended count is 0, restart the inventory at 0. Begin with an empty sequence, so the first count is 0 (OEIS A342585; Numberphile 2022).
Each count includes all terms appended earlier in the current inventory, as well as those from previous inventories. The first inventory is therefore 0, the second is 1, 1, 0, and the third is 2, 2, 2, 0. The resulting sequence begins 0, 1, 1, 0, 2, 2, 2, 0, 3, 2, 4, 1, 1, 0, 4, 4, 4, 1, 4, 0, ... (OEIS A342585). This uses occurrence counts, whereas van Eck's sequence records the number of steps since the preceding occurrence of the current term, using 0 when there was no preceding occurrence.
