Login Failed for Domain\ServerName$ when trying to create a new SQL Server Reporting Services Application in SharePoint 2013 Basic setup:  
All of the following is installed on Domain\Servername:


SharePoint 2013 Standalone Install (Trial version)      
Currently using an instance of SQL Server 2012 SP1 Express for basic
SharePoint databases (upgraded from SQL Server 2008 R2 Evaluation).
    
Also using an instance of SQL Server 2012 SP1 Evaluation for
Reporting Services for Sharepoint (Express instance doesn't support
Reporting Services for Sharepoint Integration and Express instance
couldn't be upgraded to Evaluation)  


Problem:
Ok, so I'm going through the steps to Install Reporting Services Sharepoint Mode for Sharepoint 2013 that can be found at http://msdn.microsoft.com/en-us/library/jj219068.aspx.  When I get to the end of Step 3: Create a Reporting Services Service Application and press OK, I get the following error: Login Failed for user 'Domain\Servername$'.  
According to what I've found on the interweb this means that "a process running as NETWORK SERVICE or as LocalSystem has accessed a remote resource, has authenticated itself as the machine account and was denied authorization." 
This makes sense since the account that is used for database provisioning for the new Reporting Database that needs to be created behind the scenes as part of Step 3 is the Identity for the Application Pool of SharePoint Central Administration which I currently have set as Network Service. 
Anyhoo, the agreed upon solution is to add the login Domain\Servername$ to the SQL Server Instance that's giving me trouble and give it the right permissions ( link for similar problem: http://support.microsoft.com/kb/889646/en-us).
So, I add the login for Domain\Servername$ and give it server roles: dbcreator, public, securityadmin and sysadmin and then just to be on the safe side I give it db_owner permissions for each database in the SQL Server 2012 Evaluation instance I have.
And I still get the same error :S
Just to get some more information I look at the SQL Server Error Logs using Management Studio and I can see that ther error ID is 18456, Severity 14 State 5 and the Reason for the error is  Could not find a login matching the name provided. [CLIENT: ] .  
Question 
Why am I getting this error if I've already added the login Domain\ServerName$? Any ideas or workarounds?
Thanks!
Add BOTH Domain\Servername$ login and NT AUTHORITY\Network Service login to SQL Server instance giving you above problem and give both logins the right permissions.

If you are working in a Spanish environment or another language try adding NT AUTHORITY\Servicio de Red  or whatever the equivalent is in your language. 

For steps on adding logins see links: http://support.microsoft.com/kb/889646/en-us  and http://www.c-sharpcorner.com/UploadFile/specialhost/how-to-fix-login-failed-for-user-nt-authority-iusrs-in-share/ and https://stackoverflow.com/questions/2806438/login-failed-for-user-domain-machinename

If you have a SQL Server Instance with logins that have been working for you when accessed by application pools and the like, you might want to check which logins are not in the troublesome SQL Server instance and that you might need.

Other comments:

Ok, even though the error said: Login Failed for user 'Domain\Servername$' and I added that login to the problematic SQL Server instance per what I'd seen described on the internet, it seems that the login I ALSO needed to add was NT AUTHORITY\Network Service.

I basically decided to add any logins that were in my normal SQL Server instance that I was using for the regular SharePoint databases and that weren't in the SQL Server instance I was using for SharePoint Reporting Services to the latter. Something else that tripped me up when I was looking originally to add the NT AUTHORITY\Network Service login was that I couldn't find it. It turns out I had to add it in Spanish as NT AUTHORITY\Servicio de red, which is wierd because the app pool has its Identity in English. I guess this happens when some systems have English as default language and some others have Spanish