In base 13, we use the digits 0-9 along with A for 10, B for 11, and C for 12.
Then the number that comes after A59C₁₃ is
A59C₁₃ + 1₁₃ = A59(C + 1)₁₃
… = A59(10)₁₃
… = A5(9 + 1)0₁₃
… = A5A0₁₃
where I use parentheses to indicate we're adding C and 1 in the same digits place, then carry the 1 into the next digits place.
