When you add all the terms up from this sequence: x² + (x-1)² + 3(x-2)² + (x-3)² + (x-4)² + (x-5)² + 3(x-6)² + (x-7)² ... it will be equal to half of (x³ + x² - x) for any positive even integer x. Prove why this works.

Example: 12² + 11² + 10² + 10² + 10² + 9² +8² + 7² + 6² + 6² + 6² + 5² + 4² + 3² + 2² + 2² + 2² + 1² if x = 12.

Note: The coefficients go 1, 1, then 3, then 3 1s, then 3, then 3 1s. The coefficients go in this order, even if there are coefficients left when the sequence stops. For example, with 6, the coefficients would go 1,1,3,1,1,1.