Problem:
Score : 300 points

Problem Statement

How many integer sequences A_1,A_2,\ldots,A_N of length N satisfy all of the following conditions?

0 \leq A_i \leq 9

There exists some i such that A_i=0 holds.

There exists some i such that A_i=9 holds.

The answer can be very large, so output it modulo 10^9 + 7.

Constraints

1 \leq N \leq 10^6

N is an integer.

Input

Input is given from Standard Input in the following format:

N

Output

Print the answer modulo 10^9 + 7.

Sample Input 1

2

Sample Output 1

2

Two sequences \{0,9\} and \{9,0\} satisfy all conditions.

Sample Input 2

1

Sample Output 2

0

Sample Input 3

869121

Sample Output 3

2511445

Solution in Go (1.14.1) (judge verdict: Accepted)

package main

import (
	"bufio"
	"fmt"
	"os"
	"strconv"
)

func main() {
	N := nextInt()

	// dp[i][j] は i個の数列で状態jである個数
	// j == 0 0も9も含まない

	dp := make([][]mint, N+10)
	for i := 0; i < len(dp); i++ {
		dp[i] = make([]mint, 4)
	}
	dp[0][0] = 1
	n10 := mint(10)
	n9 := mint(9)
	n8 := mint(8)

	for i := 1; i <= N; i++ {
		dp[i][0] = dp[i-1][0].mul(n8)
		dp[i][1] = dp[i-1][1].mul(n9).add(dp[i-1][0])
		dp[i][2] = dp[i-1][2].mul(n9).add(dp[i-1][0])
		dp[i][3] = dp[i-1][3].mul(n10).add(dp[i-1][1]).add(dp[i-1][2])
	}

	fmt.Println(dp[N][3])
}

type mint int64

var mod = mint(1e9 + 7)

// add は a + bを返します
func (a mint) add(b mint) mint {
	return (a + b) % mod
}

// sub は a - bを返します
func (a mint) sub(b mint) mint {
	return (a - b + mod) % mod
}

// mul は a * bを返します
func (a mint) mul(b mint) mint {
	return (a * (b % mod)) % mod
}

// div は a/bを返します
func (a mint) div(b mint) mint {
	return a.mul(b.inv())
}

// inv は aの逆元を返します
func (a mint) inv() mint {
	// 拡張ユークリッドの互除法
	b := mod
	u := mint(1)
	v := mint(0)
	for b > 0 {
		t := a / b
		a -= t * b
		a, b = b, a
		u -= t * v
		u, v = v, u
	}
	return (u + mod) % mod
}

// pow は a ^ bを返します
func (a mint) pow(b mint) mint {
	ans := mint(1)

	for b != 0 {
		if b&1 == 1 {
			ans = ans.mul(a)
		}
		a = a.mul(a)
		b = b >> 1
	}
	return ans
}

var stdin = initStdin()

func initStdin() *bufio.Scanner {
	bufsize := 1 * 1024 * 1024 // 1 MB
	var stdin = bufio.NewScanner(os.Stdin)
	stdin.Buffer(make([]byte, bufsize), bufsize)
	stdin.Split(bufio.ScanWords)
	return stdin
}

func nextString() string {
	stdin.Scan()
	return stdin.Text()
}

func nextBytes() []byte {
	stdin.Scan()
	return stdin.Bytes()
}

func nextInt() int {
	i, _ := strconv.Atoi(nextString())
	return i
}

func nextInt64() int64 {
	i, _ := strconv.ParseInt(nextString(), 10, 64)
	return i
}

func debug(args ...interface{}) {
	fmt.Fprintln(os.Stderr, args...)
}