According to results due to Eugène Ehrhart and to our calculation method which is fully detailed in our papers and research reports, the following answers are computed:

- For all
*0 <= t <= n*, the number of simultaneous calculations at instant*t*is given by the Ehrhart polynomial:*par(n, t) = 1/8 t^2 + [1/2, 3/4] t + [3/8, 1]*where

*[1/2, 3/4] = 1/2*if*t mod 2 = 1*and*[1/2, 3/4] = 3/4*otherwise. - For all
*n <= t <= 2n*, the number of simultaneous calculations at instant*t*is given by the Ehrhart polynomial:*par(n, t) = n t - 1/2 n^2 + 1/2 n - 3/8 t^2 + [0, 1/4] t - [3/8, 1]*

- On the first validity domain
*(0 <= t <= n)*,*par(n, t)*is maximum when*t*is maximum, i.e. when*t = n*. Hence,*maxpar(n) = 1/8 n^2 + [1/2, 3/4] n + [3/8, 1]* - On the second validity domain
*(n <= t <= 2n)*, we take the derivative of*par(n, t)*with respect to*t*:*par'(n, t) = n - 3/4 t + [0, 1/4]*. Hence,*par(n, t)*increases from*t = n*to*t = (4 n - [0, 1]) / 3*, and then decreases until*t = 2 n*: it is maximum for*t = tmax = (4 n - [0, 1]) / 3*. Since*tmax*is not always integral, a case study is necessary to get the exact value of*maxpar(n) = par(n, tmax)*, checking the possible values of*floor(tmax)*and*ceiling(tmax)*. It yields the following answer:*maxpar(n) = 1/6 n^2 + 5/6 n + [1, 23/24, 1]* -
**In conclusion, the maximum parallelism of the loop nest is:***Forall n >= 0, maxpar(n) = 1/6 n^2 + 5/6 n + [1, 23/24, 1]*

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