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Problem 45


Problem 45


Triangular, pentagonal, and hexagonal

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

     
Triangle Tn=n(n+1)/2 1, 3, 6, 10, 15, …
Pentagonal Pn=n(3n−1)/2 1, 5, 12, 22, 35, …
Hexagonal Hn=n(2n−1) 1, 6, 15, 28, 45, …

It can be verified that T285 = P165 = H143 = 40755.

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


三角形数、五边形数和六角形数

三角形数、五边形数和六角形数分别由以下公式给出:

     
三角形数 Tn=n(n+1)/2 1, 3, 6, 10, 15, …
五边形数 Pn=n(3n−1)/2 1, 5, 12, 22, 35, …
六边形数 Hn=n(2n−1) 1, 6, 15, 28, 45, …

可以验证,T285 = P165 = H143 = 40755。

找出下一个同时是三角形数、五边形数和六角形数的数。