Let $s_n$ be the shortest possible side length of a square constructed from exactly $n$ squares of positive integer side lengths. If no such square exists, let $s_n = 0$.
The first few values are as follows:
n | s(n) ---+------ 1 | 1 2 | 0 3 | 0 4 | 2 5 | 0 6 | 3 7 | 4 8 | 4 9 | 3 10 | 4 11 | 5 12 | 6 13 | 4 14 | 5
If we search this Integer Sequence in an Online Encyclopedia, something very remarkable happens: there is exactly one search hit. That sequence is A300001, or in English, "Side length of the smallest equilateral triangle that can be dissected into n equilateral triangles with integer sides, or 0 if no such triangle exists."
Do my square sequence's values agree with the triangle sequence's values?
If so, why? If not, when do they first disagree?
At first I thought, maybe there's some manner of bijection between my square dissections and the triangular dissections: if you halve each subsquare along its diagonals, numerically the result should fit in the entire square halved along its own diagonal. But fitting them together requires some nonobvious geometrical fiddling, and I'm not at all confident this subobject-size-preserving bijection is well defined over all dissections. Is it?