John counts up from 1 to 13, and then immediately counts down again to 1, and then back up to 13, and so on, alternately counting up and down: What is the integer in his list?

Your question doesn't make sense as written! You must have a number such as 1,000, and the question should read: "On what number does the count of 1,000 land when counting up to 13 and down to 1" !!. Check your question carefully.