

A233276


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


16



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, 18, 20, 19, 21, 23, 27, 63, 62, 57, 61, 50, 55, 56, 60, 42, 45, 47, 51, 49, 52, 54, 59, 32, 33, 34, 36, 35, 37, 39, 43, 38, 40, 41, 44, 46, 48, 53, 58, 127, 126, 120
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OFFSET

0,3


COMMENTS

For all n, a(A000079(n)) = A000225(n+1), i.e. a(2^n) = (2^(n+1))1.
For n>=1, a(A000225(n)) = A000325(n).
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):
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 18 20 19 21 23 27
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 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.
Permutation A233278 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) = A005187(1+a(n)), a(2n+1) = A055938(a(n)).
As a composition of related permutations:
a(n) = A233278(A054429(n)).


PROG

(Scheme, with memoizing definecmacro from Antti Karttunen's IntSeqlibrary)
(definec (A233276 n) (cond ((< n 2) n) ((even? n) (A005187 (+ 1 (A233276 (/ n 2))))) (else (A055938 (A233276 (/ ( n 1) 2))))))


CROSSREFS

Inverse permutation: A233275.
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, A233277/A233278, A233279/A233280, A003188/A006068.
Sequence in context: A154448 A099896 A160679 * A304083 A276441 A153141
Adjacent sequences: A233273 A233274 A233275 * A233277 A233278 A233279


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



