Interestingly, we can also choose odd numbers so that they add up to cubes.

For instance,
1 = 1^3,
3 + 5 = 2^3,
7 + 9 + 11 = 3^3,
13 + 15 + 17 + 19 = 4^3,
etc.

The following construction is also interesting:
The sum of the integers in each row below is a cube (1^3 in the first row, 2^3 in the second, 3^3 in the third, and so on):

1
3, 5
5, 9, 13
7, 13, 19, 25
9, 17, 25, 33, 41
11, 21, 31, 41, 51, 61
13, 25, 37, 49, 61, 73, 85

Also, we can find squares in at least every second row (ie each row with an odd index):
We can regard 1 as a square. We have 9 in row 3, then 9 and 25 in row 5, then 25 and 49 in row 7. We will find 49 and 81 in row 9, if we choose 17 to be the first number in row 9, and the difference between consecutive numbers in row 9 to be 16.

It may be an interesting exercise to work out the mathematics behind this.