for $n=2k+1$, number of trapeziums is $(2k+1)\binom{k}{2}$ for $n=2k$ , k odd, number of trapeziums is $k\binom{k}{2}$ for $n=2k$ , k even, number of trapeziums is $k\binom{k}{2}-\frac k2$ note: trapezium has different meanings in US and UK. likewise, trapezoid also.