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A255566 a(0) = 0; after which, a(2n) = A255411(a(n)), a(2n+1) = A256450(a(n)).

%I A255566 #37 May 30 2016 18:22:58
%S A255566 0,1,4,2,18,6,12,3,96,24,48,8,72,15,16,5,600,120,240,30,360,56,60,10,
%T A255566 480,87,88,20,90,21,22,7,4320,720,1440,144,2160,270,288,36,2880,416,
%U A255566 420,67,432,73,66,13,3600,567,568,107,570,109,108,26,576,111,112,27,114,28,52,9,35280,5040,10080,840,15120,1584,1680,168
%N A255566 a(0) = 0; after which, a(2n) = A255411(a(n)), a(2n+1) = A256450(a(n)).
%C A255566 This sequence can be represented as a binary tree. Each left hand child is produced as A255411(n), and each right hand child as A256450(n), when parent contains n >= 1:
%C A255566                                       0
%C A255566                                       |
%C A255566                    ...................1...................
%C A255566                   4                                       2
%C A255566        18......../ \........6                  12......../ \........3
%C A255566        / \                 / \                 / \                 / \
%C A255566       /   \               /   \               /   \               /   \
%C A255566      /     \             /     \             /     \             /     \
%C A255566    96       24         48       8          72       15         16       5
%C A255566 600  120 240  30    360  56   60 10     480  87   88  20     90  21   22 7
%C A255566 etc.
%C A255566 Because all terms of A255411 are even it means that odd terms can occur only in odd positions (together with some even terms, for each one of which there is a separate infinite cycle), while terms in even positions are all even.
%C A255566 After its initial 1, A255567 seems to give all the terms like 2, 3, 12, ... where the left hand child of the right hand child is one more than the right hand child of the left hand child (as for 2: 16 = 15+1, as for 3: 22 = 21+1, as for 12: 88 = 87+1).
%H A255566 Antti Karttunen, <a href="/A255566/b255566.txt">Table of n, a(n) for n = 0..1023</a>
%H A255566 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%H A255566 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A255566 a(0) = 0; after which, a(2n) = A255411(a(n)), a(2n+1) = A256450(a(n)).
%F A255566 Other identities:
%F A255566 For all n >= 0, a(2^n) = A001563(n+1). [The leftmost branch of the binary tree is given by n*n!]
%F A255566 For all n >= 0, a(A083318(n)) = A000142(n+1). [And the next innermost vertices by (n+1)! This follows because A256450(n*n! - 1) = (n+1)! - 1.]
%F A255566 For all n >= 1, A257679(a(n)) = A001511(n).
%o A255566 (Scheme, with memoization-macro definec)
%o A255566 (definec (A255566 n) (cond ((zero? n) n) ((even? n) (A255411 (A255566 (/ n 2)))) (else (A256450 (A255566 (/ (- n 1) 2))))))
%Y A255566 Inverse: A255565.
%Y A255566 Cf. A000142, A001511, A001563, A083318, A255411, A256450, A257679.
%Y A255566 Cf. also A255567 and arrays A257503, A257505.
%Y A255566 Related or similar permutations: A273666, A273667.
%K A255566 nonn,base,tabf
%O A255566 0,3
%A A255566 _Antti Karttunen_, May 05 2015
%E A255566 Formula changed because of the changed starting offset of A256450 - _Antti Karttunen_, May 30 2016