求含模运算的递推函数Fₙ(t)的显式表达式

Let's dig into this recursive function problem you're working on. First, let's restate the core definitions clearly to ensure we're on the same page:

• Base case: $F_0(t) = t$
• Recursive step: $F_n(t) = Q(F_{n-1}(t))$, where $Q(t) = W(t \bmod H) + \lfloor \frac{t}{H} \rfloor$ (I’m assuming $\frac{t}{H}$ here refers to integer/floor division since all variables are integers—if that’s not the case, feel free to clarify!)

Breaking Down What $Q(t)$ Does #

First, let's unpack $Q(t)$ to understand its behavior. For any integer $t$, we can write it as $t = kH + r$, where $k = \lfloor \frac{t}{H} \rfloor$ (the quotient when dividing by $H$) and $r = t \bmod H$ (the remainder, so $0 \leq r < H$). Plugging this into $Q(t)$ gives:
$$Q(t) = W \cdot r + k$$

In plain terms, $Q(t)$ swaps the "weight" of the remainder and quotient: instead of having $H$ times the quotient plus the remainder, we get $W$ times the remainder plus the quotient. That modular remainder term is exactly what makes finding a closed-form expression so tricky—modular operations don’t play nicely with linear combinations when you’re trying to unroll a recurrence repeatedly.

Why a Closed-Form Is Hard to Find #

Let's expand the first few recursive steps to see how messy things get quickly:

• $F_1(t) = Q(t) = W(t \bmod H) + \lfloor \frac{t}{H} \rfloor$
• $F_2(t) = Q(F_1(t)) = W\left( \left( W(t \bmod H) + \lfloor \frac{t}{H} \rfloor \right) \bmod H \right) + \lfloor \frac{W(t \bmod H) + \lfloor \frac{t}{H} \rfloor}{H} \rfloor$

By the second iteration, the new remainder term depends on both the original remainder and quotient scaled by $W$, which creates a tangled dependency that’s hard to generalize across $n$ steps. There’s no straightforward way to "unroll" this recurrence into a simple non-recursive formula because the modular operation breaks linearity.

Investigating That Periodicity You Noticed #

You’re absolutely right that $F_n(t)$ has periodic behavior. Let’s think about what defines a period $T$: we need $F_n(t + T) = F_n(t)$ for all $t$ (and for all $n$, or at least for sufficiently large $n$).

A useful way to model this is to look at $t$ in base $H$. If we write $t$ as a base-$H$ number: $t = t_0 + t_1 H + t_2 H^2 + ... + t_m H^m$ (where each $0 \leq t_i < H$), then applying $Q(t)$ transforms this digit sequence into:
$$Q(t) = W t_0 + t_1 + t_2 H + ... + t_m H^{m-1}$$

Each application of $Q$ shifts the digits right by one (in base $H$) and multiplies the least significant digit by $W$. This is similar to a state machine where each state is the current digit sequence, so the periodicity will depend on how $W$ interacts with $H$:

• If $W$ and $H$ are coprime ($\gcd(W,H) = 1$), the period may tie back to the multiplicative order of $W$ modulo $H$ (the smallest positive integer $k$ where $W^k \equiv 1 \mod H$).
• If $W$ and $H$ share a common divisor $d = \gcd(W,H)$, you can split the problem into analyzing components modulo $d$ and $H/d$ separately.

To find the exact period, one practical approach is to track the orbit of a value under repeated $Q$ applications—since all values will eventually enter a cycle, the overall period of the function will be the least common multiple (LCM) of the lengths of these cycles.

Next Steps to Explore Further #

• Digit Representation: Work with $t$ in base $H$—modeling each $Q$ operation as a digit transformation can help you see patterns in how the recurrence evolves over $n$ steps.
• Small Test Cases: Try small values of $W$ and $H$ (e.g., $W=2, H=3$ or $W=3, H=4$) and compute $F_n(t)$ manually for a few $t$ and $n$. This often reveals patterns in periodicity that you can generalize.
• Modular Invariants: For fixed $n$, analyze $F_n(t) \mod M$ for different $M$ to find invariants or conditions that hold across all $t$, which can help narrow down periodicity.

内容的提问来源于stack exchange，提问作者pieter3d