One is an odd number
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Because they are one more than the even numbers
If at least one of the numbers is even, the result will be even. Otherwise all the numbers are odd and the result will be odd.
The parity rules are: Odd + Odd = Even Even + Even = Even Odd + Even = Odd Even + Odd = Odd So the parity where one number is even is that of the other number. This means that you can go through a list of numbers and ignore all the even numbers. Every PAIR of odd numbers has even parity and by the previous paragraph, even parity can be ignored. So you can pair off odd numbers and ignore them.
Any pair of one even and one odd number will have an even product and an odd sum.
There is no such pattern because there are no even odd numbers. Odd numbers, by definition, are odd and therefore, not even.