Squares on a chessboard, tilted ones included

MediumCombinatorics Brainteasers~6m

Mark the 81 corner points of the cells of a standard $8\times8$ chessboard (a $9\times9$ grid of points). How many squares - of any size and any orientation, tilted ones included - have all four vertices on these points?

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Every lattice square fits snugly inside an axis-aligned $s\times s$ box, with one vertex on each side of the box.
Each such box holds exactly $s$ squares, and there are $(9-s)^2$ boxes. Sum $s(9-s)^2$ for $s = 1..8$ .

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Squares on a chessboard, tilted ones included

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