Gosper's unusual formula connecting $e$ and $\pi$

Wolfram MathWorld quotes (see equation $(26)$)

Gosper gives the unusual equation connecting $\pi$ and $e$ $$\sum_{n = 1}^{\infty}\frac{1}{n^{2}}\cos\left(\frac{9}{n\pi + \sqrt{n^{2}\pi^{2} - 9}}\right) = -\frac{\pi^{2}}{12e^{3}}\tag{1}$$

I did not find any references to a paper in which Gosper proved the above formula. I am perplexed by this cosine term in the formula and it does not seem to resemble any familiar series. A direct proof of this formula would be greatly appreciated. Or if someone knows Gosper's paper where this formula is established then do provide a link to that paper.

Update: We can see that $$\frac{9}{n\pi + \sqrt{n^{2}\pi^{2} - 9}} = n\pi - \sqrt{n^{2}\pi^{2} - 9}\tag{2}$$ and noting that $\cos(n\pi - \alpha) = (-1)^{n}\cos \alpha$ we can see that the formula of Gosper can be written as $$\sum_{n = 1}^{\infty}\frac{(-1)^{n}}{n^{2}}\cos\sqrt{n^{2}\pi^{2} - 9} = -\frac{\pi^{2}}{12e^{3}}\tag{3}$$ I think that the formula given by Gosper is probably a special case of a more general formula for the sum $$\sum_{n = 1}^{\infty}\frac{(-1)^{n}}{n^{2}}\cos\sqrt{n^{2}\pi^{2} + a^{2}}\tag{4}$$ (the formula $(3)$ corresponds to $a = 3i$). Looks like Gosper is also fond of strange formulas like Ramanujan.

Solution 1:

I know this general result: $$\displaystyle \sum_{k=1}^{\infty}\frac{\cos\bigl({k\pi-\sqrt{k^2\pi^2-a^2}\,}\bigr)}{k^2}=\frac{\pi^2}{12}\,\bigl({-\cosh(a)+\frac{3}{a}\sinh(a)}\bigr) $$

and you can see this paper:On some strange summation formulas by R. William Gosper 1993 can download by:http://projecteuclid.org/euclid.ijm/1255987146