Three columns problem

Problem posed by Misha Khotyakov.

How it works

We have a matrix with three columns and an arbitrary number of rows. The first column contains the natural number 1, 2, 3, etc. The second column contains the smallas natural number that has not occured yet if we go trought the matrix from left to right and from up to down. The third column is the sum of the first two.

Problem: find a formula for the second column

What this program does

We generate the matrix by the above prescription with a big nubmer of rows and plot the result

Outcome

it seems that the ratio of the second column to the first, and the ratio of the third column to the second approach the golden ratio