OEIS sequence A400000: The city-building sequence

The city-building sequence assigns each new building the smallest height h such that it can see exactly h buildings already built. Visibility is determined by the heights of the buildings between the pair; in the associated natural visibility graph, collinear points count as visible. The sequence begins 0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, … . For a new building, the number of earlier buildings visible from height h cannot decrease as h rises: from ground level it sees at least the immediately preceding building, and from sufficient height it sees them all. This ensures the defining height exists. The sequence is unbounded and contains every nonnegative integer; moreover, the terms just after a record high follow a predictable shifted pattern.

The sequence also records graph structure: each term is the number of earlier buildings connected to that position, so the sum of the first n terms is the number of visibility-graph edges among the first n buildings. A conjecture concerns whether the largest limiting average of these terms is at most 6. Computations find a remarkably stable distribution over the first 10.46 trillion terms: heights 1 through 5 occur about 31.13%, 32.76%, 15.68%, 9.31%, and 4.28% of the time, respectively. The tallest building found among the first 10^14 terms has height 309,063, at position 4,961,932,825,110.