At Most 2 Colors

AtCoder

IDabc279_g

Time2000ms

Memory256MB

Difficulty—

There is a grid with $1 \times N$ squares, numbered $1,2,\dots,N$ from left to right. Takahashi prepared paints of $C$ colors and painted each square in one of the $C$ colors. Then, there were at most two colors in any consecutive $K$ squares. Formally, for every integer $i$ such that $1 \le i \le N-K+1$, there were at most two colors in squares $i,i+1,\dots,i+K-1$. In how many ways could Takahashi paint the squares? Since this number can be enormous, find it modulo $998244353$. ## Constraints * All values in the input are integers. * $2 \le K \le N \le 10^6$ * $1 \le C \le 10^9$ ## Input The input is given from Standard Input in the following format: $N$ $K$ $C$ [samples]

Samples

Input #1

3 3 3

Output #1

21

In this input, we have a $1 \times 3$ grid.  
Among the $27$ ways to paint the squares, there are $6$ ways to paint all squares in different colors, and the remaining $21$ ways are such that there are at most two colors in any consecutive three squares.

Input #2

10 5 2

Output #2

1024

Since $C=2$, no matter how the squares are painted, there are at most two colors in any consecutive $K$ squares.

Input #3

998 244 353

Output #3

952364159

Print the number modulo $998244353$.

API Response (JSON)

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  "problem": {
    "name": "At Most 2 Colors",
    "description": {
      "content": "There is a grid with $1 \\times N$ squares, numbered $1,2,\\dots,N$ from left to right. Takahashi prepared paints of $C$ colors and painted each square in one of the $C$ colors.   Then, there were at mo",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc279_g"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a grid with $1 \\times N$ squares, numbered $1,2,\\dots,N$ from left to right.\nTakahashi prepared paints of $C$ colors and painted each square in one of the $C$ colors.  \nThen, there were at mo...",
      "is_translate": false,
      "language": "English"
    }
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}

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