How many regions can be selected in a chess board?

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How many regions can be selected in a 8 by 8 chess board?

Definition of region: A region is a set of cells that are all connected together(by edge). i.e. a possible region:

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I want to run(or do complexity analysis for) a backtrack algorithm, and need to know the number of possible regions.

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The value you are looking for is $51016818604894742$ and it is part of OEIS sequence A059525.