import java.util.HashMap;
public class biscuits {
long count_tastiness(long x, long[] a) {
// if (x == 1) {
// return solveXIsOne(a);
// }
HashMap<Long, Long> freq = new HashMap<>();
freq.put(a[0], 1l);
return recursiveFaster(0, freq, a, x);
//return countNaive(x, a);
}
// How many diff. sums can we get by (n[i] * 2^i, where 0 <= n[i] <= a[i]).
long solveXIsOne(long[] a) {
HashMap<Long, Long> freq = new HashMap<>();
freq.put(a[0], 1l);
return recursiveForOne(0, a[0], a[0], freq, a);
//return recursiveXIsOne(0, a[0], a);
}
HashMap<String, Long> map = new HashMap<>();
String toKey(int pos, long first) {
return pos + "." + first;
}
long recursiveXIsOne(int pos, long first, long[] a) {
if (pos == a.length - 1) {
return first + 1;
}
String key = toKey(pos, first);
if (map.containsKey(key)) map.get(key);
long res = recursiveXIsOne(pos + 1, a[pos + 1] + first / 2, a);
if (first > 0) {
res += recursiveXIsOne(pos + 1, a[pos + 1] + (first - 1) / 2, a);
}
map.put(key, res);
return res;
}
long recursiveFaster(int pos, HashMap<Long, Long> freq, long[] a, long x) {
//System.out.println(pos + ": " + freq);
if (pos == a.length - 1) {
long ans = 0;
for (long y : freq.keySet()) {
ans += freq.get(y) * (y / x + 1);
}
return ans;
}
if (freq.size() > 2 * x) {
System.exit(3);
}
HashMap<Long, Long> map = new HashMap<>();
// for each y in freq.keyset(): a[pos + 1] + y/2 and (y-1)/2.
for (long y : freq.keySet()) {
long val = a[pos + 1] + y / 2;
if (!map.containsKey(val)) map.put(val, 0l);
map.put(val, map.get(val) + freq.get(y));
if (y >= x) {
val = a[pos + 1] + (y - x) / 2;
if (!map.containsKey(val)) map.put(val, 0l);
map.put(val, map.get(val) + freq.get(y));
}
}
return recursiveFaster(pos + 1, map, a, x);
}
long recursiveForOne(int pos, long l, long r, HashMap<Long, Long> freq, long[] a) {
if (pos == a.length - 1) {
long ans = 0;
for (long y = l; y <= r; y++) {
ans += freq.get(y) * (y + 1);
}
return ans;
}
// for each l <= y <= r: a[pos + 1] + y/2 and (y-1)/2
long min = Long.MAX_VALUE;
for (long y = l; y <= r; y++) {
min = Math.min(min, a[pos + 1] + y/2);
if (y > 0) {
min = Math.min(min, a[pos + 1] + (y-1)/2);
}
}
long max = Long.MIN_VALUE;
for (long y = l; y <= r; y++) {
max = Math.max(max, a[pos + 1] + y/2);
}
HashMap<Long, Long> map = new HashMap<>();
for (long y = l; y <= r; y++) {
long val = a[pos + 1] + y / 2;
if (!map.containsKey(val)) map.put(val, 0l);
map.put(val, map.get(val) + freq.getOrDefault(y, 0l));
if (y > 0) {
val = a[pos + 1] + (y - 1) / 2;
if (!map.containsKey(val)) map.put(val, 0l);
map.put(val, map.get(val) + freq.getOrDefault(y, 0l));
}
}
return recursiveForOne(pos + 1, min, max, map, a);
}
long countNaive(long x, long[] a) {
return countRec(x, a, 0);
}
long countRec(long x, long[] a, int index) {
if (index == a.length - 1) {
return a[a.length - 1] / x + 1;
}
long temp = a[index + 1];
a[index + 1] = a[index + 1] + a[index] / 2;
long answer = countRec(x, a, index + 1);
if (a[index] >= x) {
a[index + 1] = temp + (a[index] - x) / 2;
answer += countRec(x, a, index + 1);
a[index + 1] = temp;
}
a[index + 1] = temp;
return answer;
}
}
// (s[k-1] - i * X) / (2^(k-1)), 0 <= i < 2^(k - 1).
// For which i is this state valid?
// If for each position b s.t. b-th bit is set in i, (s[b+1] - (2^b + prev(i,b))X) / 2^(b+1) >= X.
