The function f(X,Y) = (X + Y)(X + Y + 1)/2, where X and Y are non-negative integers, produces exactly the set of triangular numbers (the sums of the first k positive integers for k ≥ 1). This is because the expression simplifies to the k-th triangular number where k = X + Y, and since X and Y are non-negative integers, k can be any positive integer.
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If you're someone who's looking to study maths or something similar at Oxford or CambridgeAñadido:
If you're someone who's looking to study maths or something similar at Oxford or Cambridge, I think you'll really like this problem. For a pair of integers X and Y with X at least zero and Y at least zero, we define f of XY as a half times X plus Y times X plus Y plus one plus Y. What is a set of possible values that f of XY can take? Is it all positive integers, all positive even integers, all positive integers except for odd prime numbers, all positive integers that are triangular numbers, those of which that are the sum of the first k positive integers for k at least one, or all positive integers except for the triangular numbers? A really, really nice problem. Leave your answers in the comments down below.
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