---

Problem 321

---

Swapping Counters

A horizontal row comprising of 2n + 1 squares has n red counters placed at one end and n blue counters at the other end, being separated by a single empty square in the centre. For example, when n = 3.

A counter can move from one square to the next (slide) or can jump over another counter (hop) as long as the square next to that counter is unoccupied.

Let M(n) represent the minimum number of moves/actions to completely reverse the positions of the coloured counters; that is, move all the red counters to the right and all the blue counters to the left.

It can be verified M(3) = 15, which also happens to be a triangle number.

If we create a sequence based on the values of n for which M(n) is a triangle number then the first five terms would be: 1, 3, 10, 22, and 63, and their sum would be 99.

Find the sum of the first forty terms of this sequence.

---

交换筹码

在一行2n+1个水平方格上，一端摆有n个红色筹码，另一端摆有n个蓝色筹码，中间用一个空格分开。例如，当n=3时：

筹码可以从一格移到下一格（滑动）或者在目的地为空格时跳过另一个筹码（跳跃）。

记M(n)是完全逆转彩色筹码位置所需要的最小步数，所谓完全逆转，就是说将所有的红色筹码移到右边，蓝色筹码移到左边。

可以验证M(3) = 15，恰好是一个三角形数。

如果我们将使得M(n)为三角形数的n取出来单独作为做一个数列，这个数列的前五项将会是：1、3、10、22和63。它们的和为99。

求这个数列的前40项的和。

---