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Omni-MATH / How many ways can you remove one tile from a 2014×2014 grid such that the resulting figure can be tiled by 1×3 and 3×1 rectangles?

Problem

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problem

How many ways can you remove one tile from a 2014×20142014 \times 2014 grid such that the resulting figure can be tiled by 1×31 \times 3 and 3×13 \times 1 rectangles?
Plain-text mathematical notation (without MathML)
How many ways can you remove one tile from a 2014×2014 grid such that the resulting figure can be tiled by 1×3 and 3×1 rectangles?
Original LaTeX notation
How many ways can you remove one tile from a $2014 \times 2014$ grid such that the resulting figure can be tiled by $1 \times 3$ and $3 \times 1$ rectangles?

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