

A233278


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


16



0, 1, 2, 3, 5, 4, 6, 7, 12, 10, 9, 8, 13, 11, 14, 15, 27, 23, 21, 19, 20, 18, 17, 16, 28, 25, 24, 22, 29, 26, 30, 31, 58, 53, 48, 46, 44, 41, 40, 38, 43, 39, 37, 35, 36, 34, 33, 32, 59, 54, 52, 49, 51, 47, 45, 42, 60, 56, 55, 50, 61, 57, 62, 63, 121, 113, 108
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OFFSET

0,3


COMMENTS

This permutation is obtained by "entangling" even and odd numbers with complementary pair A055938 & A005187, meaning that it can be viewed as a binary tree. Each child to the left is obtained by applying A055938(n) to the parent node containing n, and each child to the right is obtained as A005187(n+1):
0

...................1...................
2 3
5......../ \........4 6......../ \........7
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
12 10 9 8 13 11 14 15
27 23 21 19 20 18 17 16 28 25 24 22 29 26 30 31
etc.
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 zero at the root 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.
Permutation A233276 gives the mirror image of the same tree.


LINKS

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers


FORMULA

a(0)=0, a(1)=1, and thereafter, a(2n) = A055938(a(n)), a(2n+1) = A005187(1+a(n)).
As a composition of related permutations:
a(n) = A233276(A054429(n)).


PROG

(Scheme, with memoizing definecmacro from Antti Karttunen's IntSeqlibrary)
(definec (A233278 n) (cond ((< n 2) n) ((even? n) (A055938 (A233278 (/ n 2)))) (else (A005187 (+ 1 (A233278 (/ ( n 1) 2)))))))


CROSSREFS

Inverse permutation: A233277.
Cf. A005187, A054429, A055938, A256991, A256478, A256479.
Cf. also A070939 (the binary width of both n and a(n)).
Related arrays: A255555, A255557.
Similarly constructed permutation pairs: A005940/A156552, A135141/A227413, A232751/A232752, A233275/A233276, A233279/A233280, A003188/A006068.
Sequence in context: A233277 A279339 A279349 * A075168 A075169 A138606
Adjacent sequences: A233275 A233276 A233277 * A233279 A233280 A233281


KEYWORD

nonn,tabf


AUTHOR

Antti Karttunen, Dec 18 2013


EXTENSIONS

Name changed and the illustration of binary tree added by Antti Karttunen, Apr 19 2015


STATUS

approved



