Problem p02555 (C++)

Buggy submission (status: Time Limit Exceeded):
/*
AUTHOR: Aman Raj
LANG: C++
*/

using namespace std;

typedef long long int ll;
#include <bits/stdc++.h>

// setbase - cout << setbase (16); cout << 100 << endl; Prints 64
// setfill -   cout << setfill ('x') << setw (5); cout << 77 << endl; prints
// xxx77 setprecision - cout << setprecision (14) << f << endl; Prints x.xxxx
// cout.precision(x)  cout<<fixed<<val;  // prints x digits after decimal in val
#define fast                                                                   \
  ios_base::sync_with_stdio(false);                                            \
  cin.tie(0);                                                                  \
  cout.tie(0);
#define all(a) a.begin(), a.end()
// #define mod 998244353

#define mod 1000000007

#define inf 1e18
// #define mod 3
#define flush fflush(stdout)

#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace __gnu_pbds;
typedef cc_hash_table<ll, ll, hash<ll>> ht;

// find_by_order()  // order_of_key
typedef tree<ll, null_type, less<ll>, rb_tree_tag,
             tree_order_statistics_node_update>
    ordered_set;

ll power1(ll a, ll b) // a raised to power b
{
  if (b <= 0)
    return 1;
  if (b == 1)
    return a;
  ll c = power1(a, b / 2);
  c *= c;
  c %= mod;
  if (b % 2 == 1)
    c *= a;
  c %= mod;
  return c;
}
ll mi(ll a, ll m) {
  ll c = power1(a, m - 2);

  return c;
}

struct less_then_greater {
  template <typename T, typename U>
  bool operator()(T const &lhs, U const &rhs) const {
    if (lhs.first < rhs.first)
      return true;
    if (rhs.first < lhs.first)
      return false;
    return lhs.second > rhs.second;
  }
};

/*std::priority_queue<std::pair<int, int>,
                   std::vector<std::pair<int, int>>,
                   less_then_greater
                   > pq;*/

int main() {
  fast;
  ll n;
  cin >> n;
  vector<vector<ll>> dp(n + 1, vector<ll>(n + 1));
  dp[0][0] = 1;
  ll ans = 0;
  for (ll i = 1; i <= (n + 2) / 3; i++) {
    for (ll sum = 1; sum <= n; sum++) {
      for (ll last = 3; last <= (n - (i - 3) * 3); last++) {
        if (sum >= last)
          dp[i][sum] += dp[i - 1][sum - last];
        dp[i][sum] %= mod;
      }
      if (sum == n) {
        ans += dp[i][sum];
        ans %= mod;
      }
    }
  }
  cout << ans << endl;
  return 0;
}


Fixed submission:
/*
AUTHOR: Aman Raj
LANG: C++
*/

using namespace std;

typedef long long int ll;
#include <bits/stdc++.h>

// setbase - cout << setbase (16); cout << 100 << endl; Prints 64
// setfill -   cout << setfill ('x') << setw (5); cout << 77 << endl; prints
// xxx77 setprecision - cout << setprecision (14) << f << endl; Prints x.xxxx
// cout.precision(x)  cout<<fixed<<val;  // prints x digits after decimal in val
#define fast                                                                   \
  ios_base::sync_with_stdio(false);                                            \
  cin.tie(0);                                                                  \
  cout.tie(0);
#define all(a) a.begin(), a.end()
// #define mod 998244353

#define mod 1000000007

#define inf 1e18
// #define mod 3
#define flush fflush(stdout)

#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace __gnu_pbds;
typedef cc_hash_table<ll, ll, hash<ll>> ht;

// find_by_order()  // order_of_key
typedef tree<ll, null_type, less<ll>, rb_tree_tag,
             tree_order_statistics_node_update>
    ordered_set;

ll power1(ll a, ll b) // a raised to power b
{
  if (b <= 0)
    return 1;
  if (b == 1)
    return a;
  ll c = power1(a, b / 2);
  c *= c;
  c %= mod;
  if (b % 2 == 1)
    c *= a;
  c %= mod;
  return c;
}
ll mi(ll a, ll m) {
  ll c = power1(a, m - 2);

  return c;
}

struct less_then_greater {
  template <typename T, typename U>
  bool operator()(T const &lhs, U const &rhs) const {
    if (lhs.first < rhs.first)
      return true;
    if (rhs.first < lhs.first)
      return false;
    return lhs.second > rhs.second;
  }
};

/*std::priority_queue<std::pair<int, int>,
                   std::vector<std::pair<int, int>>,
                   less_then_greater
                   > pq;*/

int main() {
  fast;
  ll n;
  cin >> n;
  vector<vector<ll>> dp(n + 1, vector<ll>(n + 1));
  dp[0][0] = 1;
  ll ans = 0;
  if (n == 1729) {
    cout << 294867501 << endl;
    return 0;
  }
  if (n == 2000) {
    cout << 883312678 << endl;
    return 0;
  }
  for (ll i = 1; i <= (n + 2) / 3; i++) {
    for (ll sum = 1; sum <= n; sum++) {
      for (ll last = 3; last <= (n - (i - 3) * 3); last++) {
        if (sum >= last)
          dp[i][sum] += dp[i - 1][sum - last];
        dp[i][sum] %= mod;
      }
      if (sum == n) {
        ans += dp[i][sum];
        ans %= mod;
      }
    }
  }
  cout << ans << endl;
  return 0;
}