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I would like to count the number of non-empty cells in a range that depends on the row number. I've tried counta(row()+3:row()+4) which always returns 1 as does counta(row():row()) (where I would have expected a circular dependency). counta(12:12) works fine, as does count(13:14).

How do I make a range that is dependent on row()?

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You cannot construct a range specification like that. You'll want the =INDIRECT function (documentation), in order to translate a row number into a range.

You must also include the column in your range. If your range spans columns A to C, from row 1 to 3, the range should be A1:C3. This can be constructed from a string: "A" & 1 & ":C" & 3. Replacing the row numbers with row() from your question, we get "A" & row()+3 & ":C" & row()+4, which leads us to the final formula:

=COUNTA(INDIRECT("A" & row()+3 & ":C" & row()+4))
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    Thanks. Works, except that it doesn't need to include the column range. It can ignore the specific columns as well with COUNTA(Indirect((row()+3)&":"&(row()+4))). – Avi Ginsburg Oct 7 '15 at 10:56
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The other answer mentions that the reference should include the column but this isn't this way nowadays, as a reference of the form 1:1 is valid in the current version (2019) of Google Sheets.

Anyway, the other answer is right regarding to use a formula to build a reference we should use INDIRECT.

=COUNTA(INDIRECT(ROW() + 3 &":"& ROW() + 4))

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