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I need some kind of "autoincrement field" in my spreadsheet. I've added =ROW() formula to first column. But as I'm adding new rows - I have to apply this formula to new created rows as well. Is there any ways to do it automatically?

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Use the following formula for that:

=ARRAYFORMULA(ROW(A:A))

I strongly advise you to use hard coded Id's. If for some reason the sheet gets re-arranged, then the Id's will follow with that re-arrangement. An ARRAYFORMULA will stick to its plan and present the range as ever.

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