1,2,4,7,11,16,22,
what is the rule?

Find rates of change until you find a constant.

dy/dx=1,2,3,4,5,6

d2y/dx2=1,1,1,1,1  

So the acceleration, d2y/d2x, is constant.  This means that this is a quadratic sequence of the form a(n)=an^2+bn+c.  So we can set up a system of equations to solve for the values of a,b, and c.  Using the first three points, (1,1), (2,2), and (3,4) we have:

9a+3b+c=4, 4a+2b+c=2, and a+b+c=1  getting the differences...

5a+b=2 and 3a+b=1  and getting this difference...

2a=1, so a=1/2 making 5a+b=2 become:

2.5+b=2, so b=-1/2, making a+b+c=1 become:

1/2-1/2+c=1, so c=1  so the rule is:

a(n)=0.5x^2-0.5x+1  or if you prefer to not have decimals

a(n)=(x^2-x+2)/2