Clock Problem

So basically we have n clocks and each clock has k numbers (0 to k-1). n,k being natural numbers with n >= 2 and k >= 2.

We can only move the time on a clock by 1 forward every single move (If we move a clock by 1 when its at k-1 it goes back to 0). So basically our time on a clock is modulo k.

The question is, how many different combinations of clocks can you get without:

-Moving the time on the same clock twice in a row
Example:
If n = 3 and k = 3, we cant do:
0 0 0 -> 0 0 1 -> 0 0 2 as we moved the same clock twice in  a row. However doing,
0 0 0 -> 0 0 1 -> 0 1 1 -> 0 1 2 is aloud.

-Never having the same combinasion twice.
Example:
If n = 3 and k = 2:
0 0 0 -> 0 0 1 -> 0 1 1 -> 0 1 0 -> 0 0 0 doesnt work as we went back to 0 0 0 which is a combinasion we have already done.

We also start at 0 on each clock (it doesnt rly change the solution but just extra info).

Example:
If n = 2 and k = 2,
We have 0 0 -> 0 1 -> 1 1 -> 1 0 this is the max we can do so the answer is 4.

Answer

The problem is equivalent to the number of ways the number $k$ can be written as the sum of $n$ nonnegative integers with $2x_i \leq k+1$. This condition implies that we are not moving a click twice in a row. If the condition is violated, then $$?_i???/2?+1. (1)$$ (here $??/2?$ is the smallest integer larger than $k/2$). So the total number of combinations without moving a clock twice in a row is the same as the number of nonnegative integer solutions to $$\sum_{i=1}^n ?_?=?, x_i \geq 0 (*)$$ subject to $2?_???+1$. Apply the principle of inclusion-exclusion, where the $?$ properties to be excluded are $2?_???+2$, equivalently, $?_????/2?+1$. Without any constraints the total number of solutions of (*) is $C(?+??1, ??1)$. Considering the constraints one at a time, the number of solutions violating (1) for some $1\leq i \leq n$ is $C(??(??/2?+1)+??1, ??1)$. It is not possible to satisfy (1) more than once. Putting all this together, the total number of solutions of (*) not violating (1) is $$C(?+??1,??1)?C(?,1)C(????/2?+??2,??1)$$ $$=C(?+??1,??1)??C(??/2?+??2,??1).$$
So the total number of different combinations of clocks is
$$\sum_{l=0}^{k}\left( C(l+??1,??1)??C(?l/2?+??2,??1) \right)$$
which is the answer to your questions.


You can check that for $n=2$ and $k=2$ this gives 4.


Next I explain why the number of nonnegative solutions of (*) without any constraint is $C(?+??1,??1)$. Write the number $n+k$ as the sum of $n+k$ ones: \[n+k=1+1+1+\dots +1,\] there are $n+k-1$ $"+"$ signs above. In $C(n+k-1,n-1)$ ways we can choose $n-1$ "+" signs to remove and be left with $k-1$ "+" signs. If there are no "+" signs between numbers we just add them. Then we will get a solution to $$\sum_{i=1}^n y_i=k+n, \text{with} \ \ y_i \geq 1.     (**)$$ Subtracting $n$ from both sides and letting $x_i=y_i-1$, we get a solution to (*). So there is a one to one correspondence between solutions of (*) and (**).

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The answer is accepted.