200713: [AtCoder]ARC071 F - Infinite Sequence

Memory Limit：256 MB Time Limit：2 S
Judge Style：Text Compare Creator：
Submit：0 Solved：0

Description

Score : $1000$ points

Problem Statement

How many infinite sequences $a_1, a_2, ...$ consisting of {${1, ... ,n}$} satisfy the following conditions?

  • The $n$-th and subsequent elements are all equal. That is, if $n \leq i,j$, $a_i = a_j$.
  • For every integer $i$, the $a_i$ elements immediately following the $i$-th element are all equal. That is, if $i < j < k\leq i+a_i$, $a_j = a_k$.

Find the count modulo $10^9+7$.

Constraints

  • $1 \leq n \leq 10^6$

Input

Input is given from Standard Input in the following format:

$n$

Output

Print how many sequences satisfy the conditions, modulo $10^9+7$.

Sample Input 1

2

Sample Output 1

4

The four sequences that satisfy the conditions are:

  • $1, 1, 1, ...$
  • $1, 2, 2, ...$
  • $2, 1, 1, ...$
  • $2, 2, 2, ...$

Sample Input 2

654321

Sample Output 2

968545283

Input

题意翻译

定义 $n-$可爱序列 指无限长的由 $\{1,2...,n\}$ 组成的序列。同时 $a_1,a_2...$满足以下条件: 1.第 $n$ 个及以后的元素是相同的，即若 $\forall i,j\geq n,a_i=a_j$ 。 2.对于每个位置 $i$，紧随第 $i$ 个元素后的 $a_i$ 个元素是相同的，即若 $\forall i<j<k≤i+a_i,a_j=a_k$。 输入 $n$，请输出 $n-$可爱序列的数量 $\bmod 10^9+7$ 。 $n\leq{10^6}$。 翻译 by @皎月半洒花 。

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