Nic is investigating a family of number diagrams.

Here are the first few:

1

 

1	1
1	2

 

1	1	1
1	2	2
1	2	3

 

1	1	1	1
1	2	2	2
1	2	3	3
1	2	3	4

Nic calculates the total of the numbers in each number square, forming a sequence that starts

1, 5, 14, 30…

What will the 10^th term in the sequence be?

What about the 100^th term? Or the n^th term?

Bonus question: What has this got to do with square numbers? How is this connected to the diagrams?