Special isosceles triangles

Consider the isosceles triangle with base length, $b=16$, and legs, $L=17$.

By using the Pythagorean theorem it can be seen that the height of the triangle, $h = \sqrt{17^2-8^2} = 15$, which is one less than the base length.

With $b=272$ and $L=305$, we get $h=273$, which is one more than the base length, and this is the second smallest isosceles triangle with the property that $h=b\pm 1$.

Find $\sum L$ for the twelve smallest isosceles triangles for which $h=b\pm 1$ and $b$, $L$ are positive integers.