From a geometrical point of view, all the points *(i, j, k)* of *P_n* resulting in the same value of *t = t(i, j, k)*, belong to the two-parameters rational polytope:

The number of points in *T_{n, t}* is *par(n, t)*.

Giving this last parametric definition of *T_{n, t}* as an input to the parametric vertices finding program, it outputs the following results:

- if
*0 <= t <= n*, the parametric vertices are*v1 = (t, 0, 0), v2 = (t/2, t/2, 0), v3 = (t/2, 0, t/2)***(Figure 1)**. - if
*n <= t <= 2n*, the parametric vertices are*v2 = (t/2, t/2, 0), v3 = (t/2, 0, t/2), v4 = (n, t - n, 0), v5 = (n, 0, t-n)***(Figure 2)**. - otherwise,
*T_{n, t}*does not exist and*par(n, t) = 0*.

## Figure 1 | ## Figure 2 |

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