NAND repeatedly

AtCoder

IDabc310_e

Time2000ms

Memory256MB

Difficulty—

You are given a string $S$ of length $N$ consisting of `0` and `1`. It describes a length-$N$ sequence $A=(A _ 1,A _ 2,\ldots,A _ N)$. If the $i$\-th character of $S$ $(1\leq i\leq N)$ is `0`, then $A _ i=0$; if it is `1`, then $A _ i=1$. Find the following: \\\[\\sum _ {1\\leq i\\leq j\\leq N}(\\cdots((A _ i\\barwedge A _ {i+1})\\barwedge A _ {i+2})\\barwedge\\cdots\\barwedge A _ j)\\\] More formally, find $\displaystyle\sum _ {i=1} ^ {N}\sum _ {j=i} ^ Nf(i,j)$ for $f(i,j)\ (1\leq i\leq j\leq N)$ defined as follows: \\\[f(i,j)=\\left\{\\begin{matrix} A _ i&(i=j)\\\\ f(i,j-1)\\barwedge A _ j\\quad&(i\\lt j) \\end{matrix}\\right.\\\] Here, $\barwedge$, NAND, is a binary operator satisfying the following: \\\[0\\barwedge0=1,0\\barwedge1=1,1\\barwedge0=1,1\\barwedge1=0.\\\] ## Constraints * $1\leq N\leq10^6$ * $S$ is a string of length $N$ consisting of `0` and `1`. * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $S$ [samples]

Samples

Input #1

5
00110

Output #1

9

Here are the values of $f(i,j)$ for the pairs $(i,j)$ such that $1\leq i\leq j\leq N$:

*   $f(1,1)=0=0$
*   $f(1,2)=0\barwedge0=1$
*   $f(1,3)=(0\barwedge0)\barwedge1=0$
*   $f(1,4)=((0\barwedge0)\barwedge1)\barwedge1=1$
*   $f(1,5)=(((0\barwedge0)\barwedge1)\barwedge1)\barwedge0=1$
*   $f(2,2)=0=0$
*   $f(2,3)=0\barwedge1=1$
*   $f(2,4)=(0\barwedge1)\barwedge1=0$
*   $f(2,5)=((0\barwedge1)\barwedge1)\barwedge0=1$
*   $f(3,3)=1=1$
*   $f(3,4)=1\barwedge1=0$
*   $f(3,5)=(1\barwedge1)\barwedge0=1$
*   $f(4,4)=1=1$
*   $f(4,5)=1\barwedge0=1$
*   $f(5,5)=0=0$

Their sum is $0+1+0+1+1+0+1+0+1+1+0+1+1+1+0=9$, so print $9$.
Note that $\barwedge$ does not satisfy the associative property. For instance, $(1\barwedge1)\barwedge0=0\barwedge0=1\neq0=1\barwedge1=1\barwedge(1\barwedge0)$.

Input #2

30
101010000100101011010011000010

Output #2

API Response (JSON)

{
  "problem": {
    "name": "NAND repeatedly",
    "description": {
      "content": "You are given a string $S$ of length $N$ consisting of `0` and `1`. It describes a length-$N$ sequence $A=(A _ 1,A _ 2,\\ldots,A _ N)$. If the $i$\\-th character of $S$ $(1\\leq i\\leq N)$ is `0`, then $A",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc310_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a string $S$ of length $N$ consisting of `0` and `1`. It describes a length-$N$ sequence $A=(A _ 1,A _ 2,\\ldots,A _ N)$. If the $i$\\-th character of $S$ $(1\\leq i\\leq N)$ is `0`, then $A...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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