Conjecture 1: This is an involution (self-inverse permutation) of nonnegative integers. (Which would imply that both formulas given in A227368 and A227369 involving A227370 are valid).

Conjecture 2: (which would automatically imply the conjecture 1): the only transpositions (used to compose the permutation) are of adjacent terms 2k-1 and 2k, where A061887 gives the values of k. This is true at least for the first 35 transpositions (up to k=60).

See the example section of A227368 to get a grasp of the problem.