Answer:
This is interesting but I don't think it's provable (unless it's just the square of odd numbers).
3² = 9, which is 1 more than a multiple of 8 (8)
4² = 16, which is not 1 more than a multiple of 8 (8)
5² = 25, which is 1 more than a multiple of 8 (24)
6² = 36, which is not 1 more than a multiple of 8 (32)
7² = 49, which is 1 more than a multiple of 8 (48)
8² = 64, which is not 1 more than a multiple of 8 (64)
9² = 81, which is 1 more than a multiple of 8 (80)
10² = 100, which is not 1 more than a multiple of 8 (96)