# [seqfan] Re: Anyone recognize this matrix?

Richard J. Mathar mathar at mpia-hd.mpg.de
Wed Apr 21 14:47:35 CEST 2021

```A formal description of this infinite array of 0's and 1's is:
The "full" array including a leading row of all-0 starts as follows:

0 00000000000000000
1 01010101010101010
2 01000100010001000
3 00010001000100010
4 00010000000100000
5 01000101010001010
6 01010100010101000
7 00000001000000010
8 00000001000000000
9 01010100010101010
10 01000101010001000
11 00010000000100010
12 00010001000100000
13 01000100010001010
14 01010101010101000
15 00000000000000010
16 00000000000000010
17 01010101010101000
18 01000100010001010
19 00010001000100000
20 00010000000100010
21 01000101010001000
22 01010100010101010
23 00000001000000000
24 00000001000000010
25 01010100010101000
26 01000101010001010
27 00010000000100000
28 00010001000100010
29 01000100010001000
30 01010101010101010
31 00000000000000000

Because each second column contains only zeros, we delete each second column
and get the "reduced" array

0 00000000000000000
1 11111111111111111
2 10101010101010101
3 01010101010101010
4 01000100010001000
5 10111011101110111
6 11101110111011101
7 00010001000100010
8 00010000000100000
9 11101111111011111
10 10111010101110101
11 01000101010001010
12 01010100010101000
13 10101011101010111
14 11111110111111101
15 00000001000000010
16 00000001000000000
17 11111110111111111
18 10101011101010101
19 01010100010101010
20 01000101010001000
21 10111010101110111
22 11101111111011101
23 00010000000100010
24 00010001000100000
25 11101110111011111
26 10111011101110101
27 01000100010001010
28 01010101010101000
29 10101010101010111
30 11111111111111101
31 00000000000000010

Each odd-numbered row is the binary complement of its preceding row, so
define a "depleted reduced" array just containing rows 0,2,4,6,8,...:

0 00000000000000000

2 10101010101010101

4 01000100010001000
6 11101110111011101

8 00010000000100000
10 10111010101110101
12 01010100010101000
14 11111110111111101

16 00000001000000000
18 10101011101010101
20 01000101010001000
22 11101111111011101
24 00010001000100000
26 10111011101110101
28 01010101010101000
30 11111111111111101

The definition of this seems to be given by reading the 1st, 2nd, 3rd
... column downwards, which gives periodic patterns of zeros and ones:

0,1 (col 1)
0,0,1,1 (col 2)
0,1 (col 3)
0,0,0,0,1,1,1,1 (col 4)
0,1 (col 5)
0,0,1,1 (col 6)
0,1 (col 7)
0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1 (col 8)
0,1 (col 9)
0,0,1,1 (col 10)
0,1 (col 11)
0,0,0,0,1,1,1,1 (col 12)
0,1 (col 13)
0,0,1,1 (col 14)
0,1 (col 15)
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, (col 16)

where the number of zeros (and number of ones) in the periods
of column k is given by A006519(k).

```