fractal (?) sequence with "known" fixed points

[wouter meeussen]

I got a hint from a friend (thanks Ralf) that all this boils down to
2^k+1 and 3*2^k+1 interleaved as in A029744+1
(Necklaces with n beads that are the same when turned over).
but with the first few elements not quite matching.

hmm,  pseudo-fractal permutations over the integers, necklaces,
and the slightly silly
"Values of n such that Sum[-(-1)^(k) n/k (n-1)/(k+1), {k,1,n}] (n!!) is an
integer",

I feel "turned over" a bit myself. Out of my depth (too shallow).

Wouter.