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 i

^{th}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 0

^{th}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?