fractal (?) sequence with "known" fixed points

[wouter meeussen]

1, 2, 3, 4, 5, 7, 9, 13, 17, 25, 33, 49, 65, 97, 129, 193, 257, 385, 513,
769, 1025, 1537, 2049, 3073, 4097, 6145, 8193, 12289, 16385, 24577

are fixed points of the fractal (?) sequence generated by

Nest[Flatten[{#, (Length[#] + RotateRight @ Range[Length[#], 1, -1])[[#]]}]
&, {1}, 15];
(a variant on Gray code A003100)
or 1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 15, 14, 13, 10, 11, 12, 17, 32, 31, 30,
29,
26, 27, 28, 25, 18, 19, 20, 21, 24, 23, 22, ...

the fixed points closely resemble
 A078621
Sequence :  1,5,9,11,13,17,25,33,49,65,97,129,193,257,385,513,769,1025,
1537,2049,3073,4097,6145,8193,12289,16385,24577 ..

but the logic escapes me.

Any help out there?

W.