Problem 710
One Million Members
On Sunday 5 April 2020 the Project Euler membership first exceeded one million members. We would like to present this problem to celebrate that milestone. Thank you to everyone for being a part of Project Euler.
The number $6$ can be written as a palindromic sum in exactly eight different ways:
$$(1, 1, 1, 1, 1, 1), (1, 1, 2, 1, 1), (1, 2, 2, 1), (1, 4, 1), (2, 1, 1, 2), (2, 2, 2), (3, 3), (6)$$
We shall define a twopal to be a palindromic tuple having at least one element with a value of $2$. It should also be noted that elements are not restricted to single digits. For example, $(3, 2, 13, 6, 13, 2, 3)$ is a valid twopal.
If we let $t(n)$ be the number of twopals whose elements sum to $n$, then it can be seen that $t(6) = 4$:
$$(1, 1, 2, 1, 1), (1, 2, 2, 1), (2, 1, 1, 2), (2, 2, 2)$$
Similarly, $t(20) = 824$.
In searching for the answer to the ultimate question of life, the universe, and everything, it can be verified that $t(42) = 1999923$, which happens to be the first value of $t(n)$ that exceeds one million.
However, your challenge to the “ultimatest” question of life, the universe, and everything is to find the least value of $n \gt 42$ such that $t(n)$ is divisible by one million.
一百万名用户
2020年4月5日星期日，欧拉计划的用户突破了一百万人。我们设计了本题以纪念这一里程碑。感谢各位参与欧拉计划！
将$6$写成一个回文数组的元素之和，共有八种不同的方式：
$$(1, 1, 1, 1, 1, 1), (1, 1, 2, 1, 1), (1, 2, 2, 1), (1, 4, 1), (2, 1, 1, 2), (2, 2, 2), (3, 3), (6)$$
如果一个回文数组中至少有一个元素$2$，则称之为二回数组。注意数组的元素并未被限定为一位数，例如，$(3, 2, 13, 6, 13, 2, 3)$也是一个合法的二回数组。
记$t(n)$为元素之和为$n$的二回数组的数目，可以看出$t(6) = 4$：
$$(1, 1, 2, 1, 1), (1, 2, 2, 1), (2, 1, 1, 2), (2, 2, 2)$$
类似地，$t(20) = 824$。
在寻找生命、宇宙以及任何事情的终极答案的过程中，我们发现$t(42) = 1999923$，这恰好是第一个超过一百万的$t(n)$。
你的新挑战是，寻找生命、宇宙以及任何事情的“最终极”答案：在$n\gt 42$中，找出最小的使得$t(n)$被一百万整除的数$n$。