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In this paper, we determine some new bounds for the spectral radius of a nonnegative matrix with respect to a new defined quantity, which can be considered as an average of average 2-row sums. The new formulas extend previous results using the row sums and the average 2-row sums of a nonnegative matrix. We also characterize the equality cases of the bounds if the matrix is irreducible and we provide illustrative examples comparing with the existing bounds.