The double summation in the general quadratic form

Question

According to my book and Wikipedia, a quadratic form on $\mathbb{R}^k$ is a real-valued function of the form $Q(x_1,...,x_k)=\sum_{i,j=1}^{k} a_{ij}x_ix_j.$ When I try to use this to check the general quadratic form in three variables, using the fact that A is a symmetric matrix, I obtain $$\sum_{i,j=1}^{3} a_{ij}x_ix_j=\sum_{i=1}^3\sum_{j=1}^{3} a_{ij}x_ix_j = a_{11}x_1^2+2a_{12}x_1x_2+2a_{13}x_1x_3+2a_{23}x_2x_3+a_{22}x_2^2+a_{33}x_3^2$$ My book however omit the multiplications by 2 of the three terms that have those. What am I missing here?