I forgot my lockscreen passcode, and update 6.1.3 is installed on my computer.

Locked Out, Forgot Lock or Restrictions Passcode, or Need to Restore Your Device: Several Alternative Solutions

1. iOS- Forgotten passcode or device disabled after entering wrong passcode

2. iPhone, iPad, iPod touch: Wrong passcode results in red disabled screen

3. Restoring iPod touch after forgotten passcode

4. What to Do If You've Forgotten Your iPhone's Passcode

5. iOS- Understanding passcodes

6. iTunes 10 for Mac- Update and restore software on iPod, iPhone, or iPad

Forgotten Restrictions Passcode Help

You will need to restore your device as New to remove a Restrictions passcode. Go through the normal process to restore your device, but when you see the options to restore as New or from a backup, be sure to choose New.

Also, see iTunes- Restoring iOS software.

You need to palce the device in Recovery Mode and then connect to your computer and restore via iTunes. Did you do that? The iPod will be erased.

iOS: Wrong passcode results in red disabled screen

If recovery mode does not work try DFU mode.

How to put iPod touch / iPhone into DFU mode « Karthik's scribblings

I don't believe you are reading the links provided. The first one is specifically for the problem you described, so it would make no sense that you would need a passcode you cannot remember. The second link is the same link referred to by lllaass.

You cannot save your content as it is. Fixing the problem requires that the content be erased first. Once iOS is restored you have the option of restoring as NEW or restoring a backup. But you must have a recent backup or you will lose some data.

The need to only restore as new is if you have forgotten your Restrictions passcode, not the lock screen passcode.