Telerik Web Parts for SharePoint 2013 are retired with R2 2023 being their final release. Technical support will be available until June 2024 for eligible licenses.
The SharePoint 2013 Apps are divided into several types, depending on where they are hosted. The following list contains information about each type and whether Telerik® UI for ASP.NET AJAX support it:
Provider-hosted (supported) - Developer or IT department provides the hosting structure for the application, so there is no limitation for the technology that can be used, including Telerik® UI for ASP.NET AJAX.
Autohosted (supported) - The application is hosted in Windows Azure. In general Telerik® UI for ASP.NET AJAX support Azure.
More information on SharePoint app model can be found in this MSDN article.