E. Intercity Travelling

Posted 2020-12-04 qldabiaoge

E. Intercity Travelling

time limit per test

1.5 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

Leha is planning his journey from Moscow to Saratov. He hates trains, so he has decided to get from one city to another by car.

The path from Moscow to Saratov can be represented as a straight line (well, it‘s not that straight in reality, but in this problem we will consider it to be straight), and the distance between Moscow and Saratov is $\mathrm{nn km.\; Let‘s\; say\; that\; Moscow\; is\; situated\; at\; the\; point\; with\; coordinate 00 km,\; and\; Saratov\; —\; at\; coordinate nn km.}$

Driving for a long time may be really difficult. Formally, if Leha has already covered $\mathrm{ii kilometers\; since\; he\; stopped\; to\; have\; a\; rest,\; he\; considers\; the difficulty\; of\; covering (i+1)(i+1)-th\; kilometer\; as ai+1ai+1.\; It\; is\; guaranteed\; that\; for\; every i\in [1,n?1]i\in [1,n?1] ai\le ai+1ai\le ai+1.\; The\; difficulty\; of\; the\; journey\; is\; denoted\; as\; the\; sum\; of\; difficulties\; of\; each\; kilometer\; in\; the\; journey.}$

Fortunately, there may be some rest sites between Moscow and Saratov. Every integer point from $\mathrm{11 to n?1n?1 may\; contain\; a\; rest\; site.\; When\; Leha\; enters\; a\; rest\; site,\; he\; may\; have\; a\; rest,\; and\; the\; next\; kilometer\; will\; have\; difficulty a1a1,\; the\; kilometer\; after\; it\; —\; difficulty a2a2,\; and\; so\; on.}$

For example, if $\mathrm{n=5n=5 and\; there\; is\; a\; rest\; site\; in\; coordinate 22,\; the\; difficulty\; of\; journey\; will\; be 2a1+2a2+a32a1+2a2+a3:\; the\; first\; kilometer\; will\; have\; difficulty a1a1,\; the\; second\; one\; — a2a2,\; then\; Leha\; will\; have\; a\; rest,\; and\; the\; third\; kilometer\; will\; have\; difficulty a1a1,\; the\; fourth\; — a2a2,\; and\; the\; last\; one\; — a3a3.\; Another\; example:\; if n=7n=7 and\; there\; are\; rest\; sites\; in\; coordinates 11 and 55,\; the\; difficulty\; of\; Leha‘s\; journey\; is 3a1+2a2+a3+a43a1+2a2+a3+a4.}$

Leha doesn‘t know which integer points contain rest sites. So he has to consider every possible situation. Obviously, there are ${\mathrm{2n?12n?1different\; distributions\; of\; rest\; sites\; (two\; distributions\; are\; different\; if\; there\; exists\; some\; point xx such\; that\; it\; contains\; a\; rest\; site\; in\; exactly\; one\; of\; these\; distributions).\; Leha\; considers\; all\; these\; distributions\; to\; be\; equiprobable.\; He\; wants\; to\; calculate pp —\; the\; expected\; value\; of\; difficulty\; of\; his\; journey.}}^{}$

Obviously, $\mathrm{p?2n?1p?2n?1 is\; an\; integer\; number.\; You\; have\; to\; calculate\; it\; modulo 998244353998244353.}$

Input

The first line contains one number $\mathrm{nn (1\le n\le 1061\le n\le 106)\; —\; the\; distance\; from\; Moscow\; to\; Saratov.}$

The second line contains $\mathrm{nn integer\; numbers a1a1, a2a2,\; ..., anan (1\le a1\le a2\le ?\le an\le 1061\le a1\le a2\le ?\le an\le 106),\; where aiai is\; the\; difficulty\; of ii-th\; kilometer\; after\; Leha\; has\; rested.}$

Output

Print one number — $\mathrm{p?2n?1p?2n?1,\; taken\; modulo 998244353998244353.}$

Examples

input

Copy

2
1 2

output

Copy

5

input

Copy

4
1 3 3 7

output

Copy

60

 

 理解题意题

 

　　https://www.cnblogs.com/Dillonh/p/9313493.html

公式推导过程 看这个博客

  

 1 #include <bits/stdc++.h>
 2 using namespace std;
 3 typedef long long LL;
 4 const int mod = 998244353;
 5 const LL maxn = 1e6 + 7;
 6 LL n,ans=0,a[maxn],b[maxn];
 7 int main() {
 8     b[0]=1;
 9     for (int i=1 ;i<maxn ;i++) b[i]=2*b[i-1]%mod;
10     scanf("%lld",&n);
11     for (int i=0 ;i<n ;i++) scanf("%lld",&a[i]);
12     for (int i=0 ;i<n ;i++)
13         ans=(ans+a[i]*((b[n-1-i]+((n-i-1)*b[n-1-1-i]%mod))%mod)%mod)%mod;
14     printf("%lld
",ans);
15     return 0;
16 }