#include "triples.h"
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
using vi = vector<int>;
using vvi = vector<vi>;
using vl = vector<ll>;
using vvl = vector<vl>;
using pi = pair<int, int>;
#define FOR(i, a, b) for (int i = a; i < b; i++)
#define ROF(i, a, b) for (int i = a; i >= b; i--)
#define each(a, x) for (auto &a : x)
#define all(x) (x).begin(), (x).end()
#define bg(x) (x).begin()
#define en(x) (x).end()
#define rev(x) reverse(all(x))
#define sz(x) int((x).size())
#define srt(x) sort(all(x))
#define cnt(a, x) count(all(x), a)
#define trace(x) each(a, (x)) cout << a << " "
#define mp make_pair
#define pb push_back
#define lb lower_bound
#define ub upper_bound
struct DSU {
int n; vi par, siz;
DSU() {};
DSU(int N) {
n = N; par.resize(n); siz.assign(n, 1);
iota(all(par), 0);
}
int find(int v) {
if (par[v] == v) return v;
return par[v] = find(par[v]);
}
void unite(int a, int b) {
a = find(a); b = find(b);
if (a != b) {
if (siz[a] < siz[b]) swap(a, b);
par[b] = a;
siz[a] += siz[b];
}
}
};
ll brute(vi h) {
int n = sz(h);
ll ans = 0;
// cerr << "brute function: \n";
FOR(i, 0, n) {
FOR(j, i+1, n) {
FOR(k, j+1, n) {
// vi d = {k - j, k - i, j - i}, f = {h[i], h[j], h[k]};
// srt(d); srt(f);
int d1 = min({k-j, k-i,j-i}), d3 = max({k-j, j-i, k-i});
int d2 = (k-j) ^ (k-i) ^ (j-i) ^ d1 ^ d3;
int f1 = min({h[i], h[j], h[k]}), f3 = max({h[i], h[j], h[k]});
int f2 = h[i] ^ h[j] ^ h[k] ^ f1 ^ f3;
// bool valid = d[0] == f[0] && d[1] == f[1] && d[2] == f[2];
// if (!valid) continue;
// ans++;
// cerr << i << " " << j << " " << k << "\n";
if (d1 == f1 && d2 == f2 && d3 == f3) ans++;
}
}
}
return ans;
}
struct Triple {
int i, j, k;
Triple() {};
Triple(int I, int J, int K) {
i = min({I, J, K});
k = max({I, J, K});
j = (I ^ J ^ K) ^ i ^ k; // lol
}
const bool operator<(const Triple &other) const {
if (i != other.i) return i < other.i;
if (j != other.j) return j < other.j;
return k < other.k;
}
};
ll increasing(vi h) {
int n = sz(h);
set<Triple> used;
FOR(i, 0, n) {
int k = i - h[i];
if (k < 0) continue;
int j1 = k + h[k], j2 = i - h[k];
if (k < j1 && j1 < i) {
int e = j1 + h[j1];
if (e == i) {
used.insert(Triple(i, j1, k));
}
}
if (k < j2 && j2 < i) {
int e = j2 - h[j2];
if (e == k) {
used.insert(Triple(i, j2, k));
}
}
}
// for (auto &el : used) {
// cerr << el.i << " " << el.j << " " << el.k << "\n";
// }
return sz(used);
}
ll smart(vi h) {
int n = sz(h);
ll ans = 0;
FOR(i, 0, n) { // count number of triples where h[i] = d(i, j) or d(i, k)
int H = h[i];
set<Triple> trips; // prevent double count with dupe height
int j = i + H;
if (j < n) {
int k1 = i + h[j], k2 = j + h[j]; // 2 cases — h[j] = d(i, k) or h[j] = d(j, k)
if (k1 < n && h[k1] == (k1 - j)) trips.insert(Triple(i, j, k1)); // we know d(i, j) and d(i, k) to be h[i] and h[j], so does h[k] == d(j, k)
if (k2 < n && h[k2] == (k2 - i)) trips.insert(Triple(i, j, k2)); // symmetric but this time for i
}
int k = i + H;
if (k < n) {
int j1 = i + h[k], j2 = k - h[k]; // h[k] = d(i, k), h[k] = d(j, k)
if ((i < j1 && j1 < k) && h[j1] == (k - j1)) trips.insert(Triple(i, j1, k));
if ((i < j2 && j2 < k) && h[j2] == (j2 - i)) trips.insert(Triple(i, j2, k));
}
ans += trips.size();
}
unordered_map<int, vector<int>> pc, nc; // positive line intercepts, negative line intercepts
vi inp(n), inn(n); // lookup for the intercept
FOR(i, 0, n) {
int c1 = h[i] - i, c2 = h[i] + i;
inp[i] = c1, inn[i] = c2;
pc[c1].pb(i), nc[c2].pb(i);
}
const int SQRT = 25;
unordered_map<int, vector<vector<int>>> ppref, npref;
for (auto &el : pc) {
ppref[el.first] = vector<vector<int>>(SQRT, vector<int>(n+1, 0));
npref[el.first] = vector<vector<int>>(SQRT, vector<int>(n+1, 0));
FOR(H, 1, SQRT) {
for (int i : el.second) {
if (i + H < n && inp[i+H] == inp[i] && h[i] != H && h[i+H] != H) ppref[el.first][H][i+1] = 1;
if (i + H < n && inn[i+H] == inn[i] && h[i] != H && h[i+H] != H) npref[el.first][H][i+1] = 1;
}
FOR(i, 1, n+1) ppref[el.first][H][i+1] += ppref[el.first][H][i];
FOR(i, 1, n+1) npref[el.first][H][i+1] += npref[el.first][H][i];
}
}
FOR(i, 0, n) {
int H = h[i];
int c1 = -i, c2 = inp[i];
if (H < SQRT) {
if (ppref.count(c1)) {
ans += ppref[c1][H][n] - ppref[c1][H][i+1];
}
} else {
for (auto &el : pc[c1]) {
if (el <= i) continue;
int j = el, k = el + H;
if (!(i < j && j < k && k < n)) continue;
if (inp[k] != inp[j]) continue;
if (h[j] == H || h[k] == H) continue;
ans++;
}
}
if (H < SQRT) {
if (npref.count(c2)) {
ans += npref[c2][H][n] - npref[c2][H][i+1];
}
} else {
for (auto &el : pc[c2]) {
if (el <= i) continue;
int j = el, k = el + H;
if (!(i < j && j < k && k < n)) continue;
if (inn[k] != inn[j]) continue;
if (h[j] == H || h[k] == H) continue;
ans++;
}
}
}
return ans;
}
ll count_triples(vi h) {
return smart(h);
int n = sz(h);
ll ans = 0;
set<Triple> triples;
FOR(i, 0, n) {
FOR(j, 0, i) {
// for a given pair (i, j) the possible values for k are i - h[i], i - h[j], j+h[i], j+h[j]
int dist = i - j;
vi K = {i-h[i], i-h[j], j+h[i], j+h[j]}; // there are O(4n^2) triples total??
set<int> seen;
each(k, K) {
if (!(j < k && k < i)) continue;
if (seen.count(k)) continue;
seen.insert(k);
vi d = {abs(i-j), abs(i-k), abs(j-k)}, f = {h[i], h[j], h[k]};
srt(d); srt(f);
bool valid = d[0] == f[0] && d[1] == f[1] && d[2] == f[2];
if (!valid) continue;
ans++;
triples.insert(Triple(i, j, k));
}
}
}
for (auto &el : triples) {
cerr << el.i << " " << el.j << " " << el.k << "\n";
}
return ans;
}
vi construct_range(int M, int K) {
// vi arr = {3, 1, 1, 2, 4, 3, 1, 2, 1, 4, 3, 2};
// vi ret = {};
// while (sz(ret) < M) {
// each(el, arr) ret.pb(el);
// }
// while (sz(ret) > M) ret.pop_back();
// return ret;
int best = -1;
vi bc;
int LO = 4, HI = 20, MAX_CALL = 4;
auto dfs = [&](const auto &self, vi cur) {
if (sz(cur) == M) {
ll tl = brute(cur);
if (tl > best) {
cerr << "Found an array with " << tl << " triples.\n";
trace(cur); cerr << "\n";
best = tl;
bc = cur;
}
return;
}
FOR(i, 1, MAX_CALL + 1) {
vi ncur = cur;
ncur.pb(i);
self(self, ncur);
}
};
vi initial = {};
// dfs(dfs, initial);
int cnt = 0; bool flip = false;
for (M = LO; M <= HI; M += LO) {
dfs(dfs, initial);
initial = bc;
}
// for (auto &el : bc) {
// cerr << el << ", ";
// }
// cerr << "\n";
return bc;
}
