Closed formula for the sum $a^1+a^4+a^9...$

According to Jacobi Triple product formula we have

$$prod_{m=1}^{infty}(1-x^{2m})(1+x^{2m-1}y^2)left(1+frac{x^{2m-1}}{y^2}right)=sum_{n=-infty}^{infty}x^{n^2}y^{2n}$$ for $|x|<1$ and $yneq0$. Then taking $x=a$ and $y=1$ we get $$prod_{m=1}^{infty}(1-a^{2m})(1+a^{2m-1}1^2)left(1+frac{x^{2m-1}}{1^2}right)=sum_{n=-infty}^{infty}a^{n^2}1^{2n}$$

$$impliesprod_{m=1}^{infty}(1-a^{2m})(1+a^{2m-1})^2=2sum_{n=1}^{infty}a^{n^2}+1$$

$$impliessum_{n=1}^{infty}a^{n^2}=frac{prod_{m=1}^{infty}(1-a^{2m})(1+a^{2m-1})^2-1}{2}$$

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