[seqfan] Re: Anyone recognize this matrix?

[Neil Sloane]

Richard,  You are right, and indeed one can say much more. In fact this is
part of a bigger investigation and there is a paper in progress that will
reveal everything. I was hoping to have it finished a week ago, but keeping
the OEIS running takes a great deal of time.  Once the paper is in readable
form I will post a link to it here.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Wed, Apr 21, 2021 at 8:47 AM Richard J. Mathar <mathar at mpia-hd.mpg.de>
wrote:

> 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).
>
> --
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>