There are two issues here.
First, on iOS 7 (which your profile says you are using), double-clicking the Home button does not enable the function to delete apps. It opens the multi-tasking pane to close apps that are open. To delete apps, you tap and hold an icon on the home page until the X appears. Tapping that X of course removes the app from the device.
You can check to see if the app hasn't accidentally been moved to a folder. Swipe down anywhere on the iPad home screen to display Spotlight search. Type the app name. If it appears, it is somewhere on the iPad. If not, and the App Store still says it is installed, then try resetting your iPad following the instructions below:
Reset your iPad by simultaneously pressing and holding the Home and Sleep/Wake buttons until you see the Apple Logo. This can take up to 15 seconds so be patient and don't release the buttons until the logo appears.
Try again to see if the problem persists.
The second problem is that if you bought an app directly on the iPad, it will not appear in iTunes until you transfer purchases. In iTunes with your device connected, click File > Devices > Transfer Purchase from <your device name>.