[seqfan] An astonishing regularity with the participation of Thue-Morse sequence

[Vladimir Shevelev]

Dear SeqFans,

I have just submitted two new sequences A268865 
and A268866.
Let b(n) be the parity of number of 1's in the 
n-th term of A039724(n) {0,1,0,1,1,0,1,0,0,...}.
Let A_n be Thue-Morse sequence (A010060) 
beginning with A010060(n). Then for A268865 
a(n) is defined as the maximal number of the first
terms of {b(n)} coinciding with the corresponding 
terms of A_n. A268865 begins (n>=0):
2, 0, 0, 3, 0, 1, 2, 0, 0, 1, 22
I supposed that a(10) =22 is somewhat "resonance".
The sequence A268866 lists the records of A268865.
And here, after Peter's calculations, I was surprised
by a nice regularity. Here a(n)=A268866.
Conjecture 1) To every a(n) corresponds a position 
m=m(n) in A268865 such that either 
a) m == 0 (mod 10) or b) m=3 or m == 8 (mod 10) 
such that we have {m(n)}={0,3,10,58,170,938,2730,
15018,43690,...}; 
2) In case a) a(n)=2*m+2; in case b) a(n)=(7*m+12)/11.
It is a rare event when for records of a non-simple sequence
there appear so remarkable regularities!

All comments are welcome.

Best regards,
Vladimir