一个汉诺塔问题

[K行天下]

A Double Tower of Hanoi contains twice the number of disk as the regular Tower of Hanoi
problem, where each disk size appears twice. So there are 2n disks, of n different sizes (n>0). As
usual, there are 3 pegs. The objective is to transfer the whole tower form the original peg to one of
the other two pegs, moving only one disk at a time, without putting a larger one over a smaller one.
Putting a same-sized disk onto another is okay. If we are required to reproduce the original
top-to-bottom order arrangement, how many moves (minimal) does it take? Remember, disks of
equal size need to be in original order, and cannot be inversed.

[sagegz]

LZ好久不见!

[大王派我去巡山]

这题特殊的一点在于要求最后尺寸相同的盘子不能倒置。
用f(n)来表示移动2n个这样的盘子(最后尺寸相同的盘子没有倒置)需要的最少移动次数，
用g(n)来表示移动2n个这样的盘子(最后尺寸相同的盘子全部倒置)的最少移动次数。

边界条件：
f(1)=3，g(1)=2
状态转移方程：
g(n)=f(n-1)+g(n-1)+2
f(n)=4*min{f(n-1),g(n-1)}+3

[大王派我去巡山]

考虑不周，上面的转移方程是错的~

[大王派我去巡山]

用f(n)来表示移动2n个这样的盘子(没有倒置)需要的最少移动次数，
用g(n)来表示移动2n个这样的盘子(只有最大的两个同尺寸盘子倒置)的最少移动次数。
显然g(n)<f(n)

边界条件：
f(1)=3，g(1)=2
状态转移方程：
g(n)=2*g(n-1)+2
f(n)=4*g(n-1)+3

[大王派我去巡山]

最后f(n)就是题目的解，测试结果如下：
f(1)==3
f(2)==11
f(3)==27
f(4)==59
f(5)==123
f(6)==251
f(7)==507
f(8)==1019
f(9)==2043
f(10)==4091
f(11)==8187
f(12)==16379
f(13)==32763
f(14)==65531
f(15)==131067
f(16)==262139
f(17)==524283
f(18)==1048571
f(19)==2097147
f(20)==4194299
......