2, 0, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, ...

What are the next five numbers, and why?

Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

After row 1, row n is obtained by inserting n between every pair of consecutive numbers that sum to n.

row 1: 1 1
row 2: 1 2 1
row 3: 1 3 2 3 1
row 4: 1 4 3 2 3 4 1
row 5: 1 5 4 3 5 2 5 3 4 5 1
row 6: 1 6 5 4 3 5 2 5 3 4 5 6 1
row 7: 1 7 6 5 4 7 3 5 7 2 7 5 3 7 4 5 6 7 1

The number of numbers in each row are 2, 3, 5, 7, 11, 13, 19...

Will they always be prime?

Consider two points on parabola y=x

^{2}, (-a,a

^{2}) and (b,b

^{2}), where a and b are distinct real numbers.

If these two points are connected by a straight line, where does that line intersect the y-axis?

Inspired by an interactive sculpture at the Museum of Mathematics, NYC.