The definition given there is: a(0) = 0; for n > 0, a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n.

Here are the first few lines of numbers:

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 42, 17, 43, 16, 44, 15, 45, 14, 46, 79, 113, 78, 114, 77, 39, 78, 38, 79, 37, 80, 36, 81, 35, 82, 34, 83, 33, 84, 32, 85, 31, 86, 30, 87, 29, 88, 28, 89, 27, 90, 26, 91, 157, 224, 156, 225, 155

I think this may actually be impossible to define any other way, since the rule involves checking if one function image is negative or already in the sequence, if if it is applies a different function to get the final image. It is also interesting that it includes pretty much almost every number until very large numbers, and it took a while to discover a few that we know for suredo notappear.

Question

Why is this neither an arithmetic nor geometric sequence, or not even approaching a constant ratio like Fibonacci's sequence?