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A233276 a(0)=0, a(1)=1, after which a(2n) = A005187(1+a(n)), a(2n+1) = A055938(a(n)).

%I A233276 #25 Apr 19 2015 20:49:35
%S A233276 0,1,3,2,7,6,4,5,15,14,11,13,8,9,10,12,31,30,26,29,22,24,25,28,16,17,
%T A233276 18,20,19,21,23,27,63,62,57,61,50,55,56,60,42,45,47,51,49,52,54,59,32,
%U A233276 33,34,36,35,37,39,43,38,40,41,44,46,48,53,58,127,126,120
%N A233276 a(0)=0, a(1)=1, after which a(2n) = A005187(1+a(n)), a(2n+1) = A055938(a(n)).
%C A233276 For all n, a(A000079(n)) = A000225(n+1), i.e. a(2^n) = (2^(n+1))-1.
%C A233276 For n>=1, a(A000225(n)) = A000325(n).
%C A233276 This permutation is obtained by "entangling" even and odd numbers with complementary pair A005187 & A055938, meaning that it can be viewed as a binary tree. Each child to the left is obtained by applying A005187(n+1) to the parent node containing n, and each child to the right is obtained as A055938(n):
%C A233276                                      0
%C A233276                                      |
%C A233276                   ...................1...................
%C A233276                  3                                       2
%C A233276        7......../ \........6                   4......../ \........5
%C A233276       / \                 / \                 / \                 / \
%C A233276      /   \               /   \               /   \               /   \
%C A233276     /     \             /     \             /     \             /     \
%C A233276   15       14         11       13          8       9          10       12
%C A233276 31  30   26  29     22  24   25  28      16 17   18 20      19  21   23  27
%C A233276 etc.
%C A233276 For n >= 1, A256991(n) gives the contents of the immediate parent node of the node containing n, while A070939(n) gives the total distance to 0 from the node containing n, with A256478(n) telling how many of the terms encountered on that journey are terms of A005187 (including the penultimate 1 but not the final 0 in the count), while A256479(n) tells how many of them are terms of A055938.
%C A233276 Permutation A233278 gives the mirror image of the same tree.
%H A233276 Antti Karttunen, <a href="/A233276/b233276.txt">Table of n, a(n) for n = 0..8191</a>
%H A233276 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A233276 a(0)=0, a(1)=1, and thereafter, a(2n) = A005187(1+a(n)), a(2n+1) = A055938(a(n)).
%F A233276 As a composition of related permutations:
%F A233276 a(n) = A233278(A054429(n)).
%o A233276 (Scheme, with memoizing definec-macro from _Antti Karttunen_'s IntSeq-library)
%o A233276 (definec (A233276 n) (cond ((< n 2) n) ((even? n) (A005187 (+ 1 (A233276 (/ n 2))))) (else (A055938 (A233276 (/ (- n 1) 2))))))
%Y A233276 Inverse permutation: A233275.
%Y A233276 Cf. A005187, A054429, A055938, A256991, A256478, A256479.
%Y A233276 Cf. also A070939 (the binary width of both n and a(n)).
%Y A233276 Related arrays: A255555, A255557.
%Y A233276 Similarly constructed permutation pairs: A005940/A156552, A135141/A227413, A232751/A232752, A233277/A233278, A233279/A233280, A003188/A006068.
%K A233276 nonn,tabf
%O A233276 0,3
%A A233276 _Antti Karttunen_, Dec 18 2013
%E A233276 Name changed and the illustration of binary tree added by _Antti Karttunen_, Apr 19 2015