[seqfan] Is there an explanation for this pattern?
Ali Sada
pemd70 at yahoo.com
Sat Nov 2 08:13:54 CET 2019
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
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