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Problem 45

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Triangular, pentagonal, and hexagonal

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

$$
\begin{aligned}
&\text{Triangle} && T_n=n(n+1)/2 && 1, 3, 6, 10, 15, \ldots \\
&\text{Pentagonal} && P_n=n(3n−1)/2 && 1, 5, 12, 22, 35, \ldots \\
&\text{Hexagonal} && H_n=n(2n−1) && 1, 6, 15, 28, 45, \ldots
\end{aligned}
$$

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

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

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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：

$$
\begin{aligned}
&\text{三角形数} && T_n=n(n+1)/2 && 1, 3, 6, 10, 15, \ldots \\
&\text{五边形数} && P_n=n(3n−1)/2 && 1, 5, 12, 22, 35, \ldots \\
&\text{六边形数} && H_n=n(2n−1) && 1, 6, 15, 28, 45, \ldots
\end{aligned}
$$

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。

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