[seqfan] Is there an explanation for this pattern?

[Ali Sada]

Hi Everyone,
I wasworking on the simple array below when I saw an interesting pattern. Maybe thereis a simple explanation for it, but I couldn’t figure it out.

 
Here is the array, and I am sorry inadvance for the messy description.
 

We startwith a(1,1)=1; a(1,2)=2; a(2,1)=3; and a(2,2)=1.

 

The second numberon the diagonal is 1. It has one decimal digit. So far, there are 4 terms thathas the same number of digits in the array. So, we put a(3,3)=4. 

 

Then we fillthe third column up by adding 1 in each step: a(2,3)=5 and a(1,3)=6. 

To fill the thirdrow, we continue to the left a(3,2)=7 and a(3,1)=8.

 

Anotherexample: a(6,6)=15. It has 2 decimal digits. Up to that point (i.e. in thearray A(6,6)) there are 26 terms that also have 2 decimal digits. So, a(7,7)=26.

 

To fill the columnup we add one in each step:

a(6,7)=27;a(5,7)=28; a(4,7)=29; a(3,7)=30; a(2,7)=31; and a(1,7)=32.

 

To fill the row,we continue: a(7,6)=33; a(7,5)=34; a(7,4)=35; a(7,3)=36; a(7,2)=37; anda(7,1)=38. 

 

I uploaded animage of the array here: 

https://justpaste.it/4q98i

 

And this is thegraph of its diagonal (which I think is cool):

https://justpaste.it/70kru

 

Then Inoticed this pattern in this image (please check the terms highlighted in blue):

https://justpaste.it/73d2r

 

There are increasingperiods where a(n,n)=a(n-1,1)+1. This happens at n=2 to 3, then 6 to 10, then 19to 34, then 57 to 109, then 182 and 350, etc.
I don’t seeany direct link between a(n,n) and a(n-1,1.) I also don’t know why this happensspecifically at these periods.(When Itried the same algorithm on binary digits instead of decimal, there were similarperiods but they were shorter and more farther away from each other.)
 

As usual, Iwould really appreciate your help with this array.

 

Best,

 
Ali



1             2             6             12          14               20          32          46          62               80          100        111

3             1             5             11          13               19          31          45          61               79          99          110

8             7             4             10          12               18          30          44          60               78          98          109

15          14          13          9             11               17          29          43          59               77          97          108

18          17          16          15          10               16          28          42          58               76          96          107

25          24          23          22          21               15          27          41          57               75          95          106

38          37          36          35          34               33          26          40          56               74          94          105

53          52          51          50          49               48          47          39          55               73          93          104

70          69          68          67          66               65          64          63          54               72          92          103

89          88          87          86          85               84          83          82          81               71          91          102

110        109        108        107        106               105        104        103        102               101        90          101

122        121        120        119        118               117        116        115        114               113        112        100