Codeforces Round #372 (Div. 2) C. Plus and Square Root 数论、方程、化简、推公式

C. Plus and Square Root

time limit per test
2 seconds

memory limit per test
256 megabytes

input
standard input

output
standard output

ZS the Coder is playing a game. There is a number displayed on the screen and there are two buttons, ' + ' (plus) and '

'
(square root). Initially, the number 2 is displayed on the screen. There are n + 1 levels
in the game and ZS the Coder start at the level 1.

When ZS the Coder is at level k, he can :

Press the ' + ' button. This increases the number on the screen by exactly k.
So, if the number on the screen was x, it becomes x + k.

Press the '

'
button. Let the number on the screen be x. After pressing this button, the number becomes 

.
After that, ZS the Coder levels up, so his current level becomes k + 1. This button can only be pressed when x is
a perfect square, i.e. x = m2 for
some positive integer m.

Additionally, after each move, if ZS the Coder is at level k, and the number on the screen is m,
then m must be a multiple of k.
Note that this condition is only checked after performing the press. For example, if ZS the Coder is at level 4 and current number is 100,
he presses the '

'
button and the number turns into 10. Note that at this moment, 10 is
not divisible by 4, but this press is still valid, because after it, ZS the Coder is at level 5,
and 10 is divisible by 5.

ZS the Coder needs your help in beating the game — he wants to reach level n + 1. In other words, he needs to press the '

'
button n times. Help him determine the number of times he should press the ' + '
button before pressing the '

'
button at each level.

Please note that ZS the Coder wants to find just any sequence of presses allowing him to reach level n + 1, but not necessarily a sequence
minimizing the number of presses.

Input

The first and only line of the input contains a single integer n (1 ≤ n ≤ 100 000),
denoting that ZS the Coder wants to reach level n + 1.

Output

Print n non-negative integers, one per line. i-th
of them should be equal to the number of times that ZS the Coder needs to press the ' + ' button before pressing the '

'
button at level i.

Each number in the output should not exceed 1018.
However, the number on the screen can be greater than 1018.

It is guaranteed that at least one solution exists. If there are multiple solutions, print any of them.

Examples

input

3

output

14
16
46

input

2

output

999999999999999998
44500000000

input

4

output

217
46
97

Note

In the first sample case:

On the first level, ZS the Coder pressed the ' + ' button 14 times
(and the number on screen is initially 2), so the number became 2 + 14·1 = 16.
Then, ZS the Coder pressed the '

'
button, and the number became 

.

After that, on the second level, ZS pressed the ' + ' button 16 times,
so the number becomes 4 + 16·2 = 36. Then, ZS pressed the '

'
button, levelling up and changing the number into 

.

After that, on the third level, ZS pressed the ' + ' button 46 times,
so the number becomes 6 + 46·3 = 144. Then, ZS pressed the '

'
button, levelling up and changing the number into 

.

Note that 12 is indeed divisible by 4,
so ZS the Coder can reach level 4.

Also, note that pressing the ' + ' button 10 times
on the third level before levelling up does not work, because the number becomes 6 + 10·3 = 36, and when the '

'
button is pressed, the number becomes 

 and
ZS the Coder is at Level 4. However, 6 is
not divisible by 4 now, so this is not a valid solution.

In the second sample case:

On the first level, ZS the Coder pressed the ' + ' button 999999999999999998 times
(and the number on screen is initially 2), so the number became 2 + 999999999999999998·1 = 1018.
Then, ZS the Coder pressed the '

'
button, and the number became 

.

After that, on the second level, ZS pressed the ' + ' button 44500000000 times,
so the number becomes 109 + 44500000000·2 = 9·1010.
Then, ZS pressed the '

'
button, levelling up and changing the number into 

.

Note that 300000 is a multiple of 3,
so ZS the Coder can reach level 3.

Source

Codeforces Round #372 (Div. 2)

My Solution

数论、方程、化简、推公式

首先 当前 在 level i， res是当前屏幕上的数，i+1是目标， k是答案

则有 res + k * i = (i + 1) * (i + 1) * x *x

k * i = (i + 1) * (i + 1) * x *x - res;

这样要O(n^2)了显然不行

k=( (i + 1) * (i + 1) * x *x - res ) / i;

后来 发现 res 可以被 i 整除的， 然后就想把 i 放进去

k =  (i + 1) * (i + 1) * x *x / i - res / i;

这是终于明白了 x == i 就好 

ans[i] = k =  (i + 1) * (i + 1) * i  - res / i;

且更新 res = (i + 1) * i;

复杂度 O(n)

#include <iostream>
#include <cstdio>

using namespace std;
typedef long long LL;
const int maxn = 1e5 + 8;

LL ans[maxn];

int main()
{
#ifdef LOCAL
freopen("c.txt", "r", stdin);
freopen("c.out", "w", stdout);
int T = 1;
while(T--){
#endif // LOCAL
ios::sync_with_stdio(false); cin.tie(0);

LL res = 2;
for(LL i = 1; i < maxn; i++){
if(((i + 1)*(i + 1) - res) > 0 && ((i + 1)*(i + 1) - res) % i == 0){
ans[i] = ((i + 1)*(i + 1) - res) / i;
res = i + 1;
}
else{
ans[i] = (i + 1) * (i + 1) * i - res / i;
res = (i + 1) * i;
//cout << res << endl;
}

}
//cout << endl;

int n;
cin >> n;
for(int i = 1; i <= n; i++) cout << ans[i] << "\n";

#ifdef LOCAL
cout << endl;
}
#endif // LOCAL
return 0;
}

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