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The two ones at the start of the parent sequence represent parent and child 1-tuples in the grandparent sequence [(1) and (2) respectively], hence this sequence also starts with 1, 1 rather than 2, which would otherwise be a more sensible way to describe the pair of ones.

All other elements are effectively run-lengths of strings of the same integer in A229895.

The first occurrence of an integer, n, in the parent sequence, is the first of a run of n^n elements of value n. For later occurrences, the run length is n^k-(n-1)^k where k is the size of the k-tuple in the grandparent sequence, A229873.

The elements can be arranged into a triangle thus:

.... 1

.... 1, 4

.... 1, 5, 27

.... 1, 7, 37, 256

.... 1, 9, 61, 369, 3125

.... etc.

where the n-th line is:

.... n^1-(n-1)^1, n^2-(n-1)^2, ..., n^(k-1)-(n-1)^(k-1), n^n; 1 <= k < n

The first terms, for sufficiently large n simplifying as:

.... 1, 2n-1, 3n^2-3n+1, 4n^3-6n^2+4n-1, etc.

Row sums are first differences of A031972, and thus the cumulative sum of rows at the end of each row is A031972 itself, i.e., n*(n^n - 1)/(n-1).

The sequence pattern is an integer, n, followed by all k-tuples containing n, then (k+1)-tuples, etc., up to the n-tuples that have not yet appeared in the sequence. Directly before the integer n+1, therefore, we find the first occurrence of n^n n-tuples which contain the n^n permutations of 1 to n in lexicographic order. The cases n = 1 and n = 2 are degenerate as no tuples precede them; 1 is followed not by a tuple, but by 2, and 2 is followed by the tuple (1, 1), rather than (1, n) as with all other integers.

k-tuple clusters later in the sequence (k

Essentially, at each stage an n-hypercube of elements of size n is completed for each dimension up to the (n-1)-th, building on previous occurrences of the dimension, and then a hypercube for dimension n is begun to be built upon later.

Tuple sizes are in A229895.