[seqfan] Anyone recognize this matrix?

[Neil Sloane]

Dear Sequence Fans, I have an infinite 0,1 matrix. The first row is 01
repeated, the second row is 0100 repeated, and so on. Here are the first 32
rows.
I have a feeling I've seen this before, but I can't remember where.
I have the definition, but I would like a simple description.
Does anyone recognize this?

There are some obvious properties. In rows 8 through 15, for instance, the
mod 2 sums row 8 + row 15 = row 9 + row 14 = ... = row 11 + row 12 =
0000000100000001.
And similarly for rows 2 to 3; 4 to 7; 16 to 31; etc.


 1: 01*
 2: 0100*
 3: 0001*
 4: 00010000*
 5: 01000101*
 6: 01010100*
 7: 00000001*
 8: 0000000100000000*
 9: 0101010001010101*
10: 0100010101000100*
11: 0001000000010001*
12: 0001000100010000*
13: 0100010001000101*
14: 0101010101010100*
15: 0000000000000001*
16: 00000000000000010000000000000000*
17: 01010101010101000101010101010101*
18: 01000100010001010100010001000100*
19: 00010001000100000001000100010001*
20: 00010000000100010001000000010000*
21: 01000101010001000100010101000101*
22: 01010100010101010101010001010100*
23: 00000001000000000000000100000001*
24: 00000001000000010000000100000000*
25: 01010100010101000101010001010101*
26: 01000101010001010100010101000100*
27: 00010000000100000001000000010001*
28: 00010001000100010001000100010000*
29: 01000100010001000100010001000101*
30: 01010101010101010101010101010100*
31: 00000000000000000000000000000001*

[These are actually the odd-numbered rows 1,3,5,7,... of the matrix. The
even-numbered rows have a simple formula. Row 2k is 0^(2^m) 1^(2^m)
repeated, where m is the number of times 2 divides 2k.

Row 24 for example (where m=3) is 0000000011111111 repeated. I'm hoping for
something similar for the odd-numbered rows.]