Xor Sum 2

AtCoder

IDarc098_b

Time2000ms

Memory256MB

Difficulty—

There is an integer sequence $A$ of length $N$. Find the number of the pairs of integers $l$ and $r$ ($1 \leq l \leq r \leq N$) that satisfy the following condition: * $A_l\ xor\ A_{l+1}\ xor\ ...\ xor\ A_r = A_l\ +\ A_{l+1}\ +\ ...\ +\ A_r$ Here, $xor$ denotes the bitwise exclusive OR. Definition of XOR The XOR of integers $c_1, c_2, ..., c_m$ is defined as follows: * Let the XOR be $X$. In the binary representation of $X$, the digit in the $2^k$'s place ($0 \leq k$; $k$ is an integer) is $1$ if there are an odd number of integers among $c_1, c_2, ...c_m$ whose binary representation has $1$ in the $2^k$'s place, and $0$ if that number is even. For example, let us compute the XOR of $3$ and $5$. The binary representation of $3$ is $011$, and the binary representation of $5$ is $101$, thus the XOR has the binary representation $110$, that is, the XOR is $6$. ## Constraints * $1 \leq N \leq 2 \times 10^5$ * $0 \leq A_i < 2^{20}$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $...$ $A_N$ [samples]

Samples

Input #1

4
2 5 4 6

Output #1

5

$(l,r)=(1,1),(2,2),(3,3),(4,4)$ clearly satisfy the condition. $(l,r)=(1,2)$ also satisfies the condition, since $A_1\ xor\ A_2 = A_1\ +\ A_2 = 7$. There are no other pairs that satisfy the condition, so the answer is $5$.

Input #2

9
0 0 0 0 0 0 0 0 0

Output #2

Input #3

19
885 8 1 128 83 32 256 206 639 16 4 128 689 32 8 64 885 969 1

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Xor Sum 2",
    "description": {
      "content": "There is an integer sequence $A$ of length $N$. Find the number of the pairs of integers $l$ and $r$ ($1 \\leq l \\leq r \\leq N$) that satisfy the following condition: *   $A_l\\ xor\\ A_{l+1}\\ xor\\ ...\\",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc098_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is an integer sequence $A$ of length $N$.\nFind the number of the pairs of integers $l$ and $r$ ($1 \\leq l \\leq r \\leq N$) that satisfy the following condition:\n\n*   $A_l\\ xor\\ A_{l+1}\\ xor\\ ...\\...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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