Problem 411
Uphill paths
Let n be a positive integer. Suppose there are stations at the coordinates (x, y) = (2i mod n, 3i mod n) for 0 ≤ i ≤ 2n. We will consider stations with the same coordinates as the same station.
We wish to form a path from (0, 0) to (n, n) such that the x and y coordinates never decrease.
Let S(n) be the maximum number of stations such a path can pass through.
For example, if n = 22, there are 11 distinct stations, and a valid path can pass through at most 5 stations. Therefore, S(22) = 5. The case is illustrated below, with an example of an optimal path:
It can also be verified that S(123) = 14 and S(10000) = 48.
Find ∑ S(k5) for 1 ≤ k ≤ 30.
上行路径
记n为正整数。在所有坐标为(x, y) = (2i mod n, 3i mod n)的位置安设站点，其中0 ≤ i ≤ 2n。如果多个站点有相同坐标，我们就合并为同一个站点。
我们希望找到一条从(0, 0)到(n, n)的路径，路径上x和y坐标单调不减。
记S(n)是这样一条路径最多能够穿过的站点数目。
例如，当n = 22时，一共有11个不同的站点，一条可行路径最多能穿过5个站点，因此S(22) = 5。如下图所示是其中一条最优的可行路径：
同样可以验证S(123) = 14以及S(10000) = 48。
对于1 ≤ k ≤ 30，求∑ S(k5)。