A Piece of P.I.E

A \( 4 \times 4 \) square is made up of 16 \( 1 \times 1 \) squares, each of which is coloured either black or white. Let \(n\) be the number of these \( 4 \times 4 \) squares such that within the \( 4 \times 4 \) square there are no \( 3 \times 3 \) squares made up entirely of white \( 1 \times 1 \) squares. What is the digit sum of \(n?\)

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