0 1 1 2 3 5 8 13 ... 1 3 4 7 11 18 29 47 ... 2 4 6 10 16 26 42 68 ... ...etc...Notice that each row above is fibonacci in that, except for
the first two columns, a number in the ith column is the sum
of the numbers in the (i-1)st and (i-2)nd columns.
Also notice that the first column is just the one-up integers
starting at 0.
Here's how to generate the rest of the array:
Label the columns -1,0,1,2,3,... The first row is the usual
fibonacci sequence. We will genetate the rows consecutively
beginning 1, 2, 3, etc. For the row beginning with N, we
look for N in columns labeled with positive integers and
in the rows above the one we are generating. When we find
N in this manner, we take the number immediately to its right,
add 1 to it, and put this next to N in the line we are generating.
Then we generate the rest of the row using the fibonacci rule
For example, to genetate the row beginning with 1, we find
the 1 in the column labeled 1 and take the 2 immediately to
the right of it, add 1 to it to get 3. That 3 becomes the
number next to the 1 in the line we are generating.
Another example: The row which begins 3. We find a 3 in the
top row with a 5 next to it. We add 1 to the 5 to get the 6
as the element in the 0th column of the row beginning with 3.
Using the fibonacci rule, we find that row to be:
3 6 9 15 24 39 63 102 ...The claim was made that every positive integer can be found in
one, and only one, positive integer labeled column and
one, and only one, row of this array. Can you prove this?