A double series for $\pi$ using Fourier series and the Grothendieck-Krivine constant

We provide a double-series formula for $\pi$ obtained using the Fourier series expansion of $1\mathrm{/}\cos (x\mathrm{/}4)$ and applying the Parseval-Plancherel identity. We show that such a formula involves the Grothendieck-Krivine constant, and that the latter can therefore be expressed as a double series as well.