Although the Constitution as it stood in 1841 was unclear on the details of succession in the event of the President's death (some felt that the VP could be nothing more than a caretaker for the remainder of the term), Tyler announced to Congress that he was indeed the President and moved into the White House. He took a second oath of office to clinch the claim.
Congress went along with the bold move, though there was no firm legal basis for calling him "President Tyler" until the 25th Amendment passed in 1967. Some wags referred to him as His Accidency.
What's more, Tyler had no vice president at all during his term, which lasted almost four years. The Executive has been VP-less seventeen times.