There are many different patterns with numbers that can be figured out usually by using operation in math. These operations include multiplying, dividing, subtracting or adding. In this case, the number five hundred sixteen is the first number and then the second number is four hundred ninety-seven.
If you subtract the second number from the opening number, the remainder is nineteen. If you do the same thing for the next two numbers, the rest is also nineteen. Therefore, it seems like the pattern is nineteen less than the amount before. However, to be sure, you should check out all of the numbers. Therefore, the next number in the series would be four hundred forty since it is nineteen less than the amount before it.