Interesting.

Group 1 : 1 Sum 1 : 1

Group 2 : 3,5 Sum 2 : 8

Group 3 : 7, 9,11 Sum 3 : 27

Group 4 : 13,15,17,19 Sum 4 : 64

etc

Each group is an arithmetic progression.

If the group number is x, then the first number in the group is x(x-1)+1

Use the sum of an arithmetic progression formula with a=x(x-1)+1, d=2 and n = x (seeing that the group number is the same as the number of terms in the group)

Then do some algebra. Falls out nicely!

(Not sure if this constitutes a "real" proof as a=x(x-1)+1 was found by observation.)