Problem 45

Triangular, Pentagonal, and Hexagonal

Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:

$\begin{aligned} Triangle & T_{n} = n (n + 1) / 2 & 1, 3, 6, 10, 15, \dots \\ Pentagonal & P_{n} = n (3 n - 1) / 2 & 1, 5, 12, 22, 35, \dots \\ Hexagonal & H_{n} = n (2 n - 1) & 1, 6, 15, 28, 45, \dots \end{aligned}$

It can be verified that $T_{285} = P_{165} = H_{143} = 40755$ .

Find the next triangle number that is also pentagonal and hexagonal.

三角形数、五边形数和六边形数

三角形数、五边形数和六边形数分别由以下公式给出：

$\begin{aligned} 三角形数 & T_{n} = n (n + 1) / 2 & 1, 3, 6, 10, 15, \dots \\ 五边形数 & P_{n} = n (3 n - 1) / 2 & 1, 5, 12, 22, 35, \dots \\ 六边形数 & H_{n} = n (2 n - 1) & 1, 6, 15, 28, 45, \dots \end{aligned}$

可以验证， $T_{285} = P_{165} = H_{143} = 40755$ 。

找出下一个同时是三角形数、五边形数和六边形数的数。