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This has to do with when you take the starting point for the day I would assume. Also, I got the number from Wolfram:

http://www.wolframalpha.com/input/?i=May+14+1984+to+Septembe...

In the this case 'to' means 'through' and is inclusive of all the days. By adding the difference we do not twice count the number we arrived at from the first part of your calculation.

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