If the coefficient of f in $(1+\mathrm{x}{)}^{101}{\left(1-\mathrm{x}+{\mathrm{x}}^2\right)}^{100}$ it nonzero, then n cannot be of the form

We have,

$\begin{array}{l}(1+\mathrm{x}{)}^{101}{\left(1-\mathrm{x}+{\mathrm{x}}^2\right)}^{100}\\ =(1+\mathrm{x}){\left((1+\mathrm{x})\left(1-\mathrm{x}+{\mathrm{x}}^2\right)\right)}^{100}\\ =(1+\mathrm{x}){\left(1+{\mathrm{x}}^3\right)}^{100}\\ =(1+\mathrm{x})\left\{{\mathrm{C}}_0+{\mathrm{C}}_1{\mathrm{x}}^3+{\mathrm{C}}_2{\mathrm{x}}^6+\dots +{\mathrm{C}}_{100}{\mathrm{x}}^{300}\right\}\\ =(1+\mathrm{x})\sum _{\mathrm{r}=0}^{\mathrm{n}}{ }^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}{\mathrm{x}}^{3\mathrm{r}}=\sum _{\mathrm{r}=0}^{\mathrm{n}}{ }^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}{\mathrm{x}}^{3\mathrm{r}}+\sum _{\mathrm{r}=0}^{\mathrm{n}}{ }^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}{\mathrm{x}}^{3\mathrm{r}+1}\end{array}$