A267112 Permutation of natural numbers: a(1) = 1; a(2n) = A087686(1+a(n)), a(2n+1) = A088359(a(n)), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001.

1, 2, 3, 4, 5, 7, 6, 8, 9, 12, 10, 15, 13, 14, 11, 16, 17, 21, 18, 27, 22, 24, 19, 31, 28, 29, 23, 30, 25, 26, 20, 32, 33, 38, 34, 48, 39, 42, 35, 58, 49, 51, 40, 54, 43, 45, 36, 63, 59, 60, 50, 61, 52, 53, 41, 62, 55, 56, 44, 57, 46, 47, 37, 64, 65, 71, 66, 86, 72, 76, 67, 106, 87, 90, 73, 96, 77, 80, 68, 121, 107, 109, 88

Offset

1,2

Comments

This sequence can be represented as a binary tree. Each left hand child is produced as A087686(1+n), and each right hand child as A088359(n), when their parent contains n:

|

...................1...................

2 3

4......../ \........5 7......../ \........6

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

8 9 12 10 15 13 14 11

16 17 21 18 27 22 24 19 31 28 29 23 30 25 26 20

etc.

The level k of the tree contains all numbers of binary width k, like many base-2 related permutations (A003188, A054429, etc). For a proof, see A267110, which gives the contents of each parent node (for node containing n).

A276442 shows the mirror-image of the same tree.

Links

Antti Karttunen, Table of n, a(n) for n = 1..8192

T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.

Index entries for Hofstadter-type sequences

Index entries for sequences that are permutations of the natural numbers

Formula

a(1) = 1; after which, a(2n) = A087686(1+a(n)), a(2n+1) = A088359(a(n)).

As a composition of other permutations:

a(n) = A276442(A054429(n)).

a(n) = A276344(A233276(n)).

a(n) = A276346(A233278(n)).

a(n) = A276446(A003188(n)).

Other identities. For all n >= 0:

a(2^n) = 2^n. [Follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper.]

a(A000225(n)) = A006127(n), i.e., a((2^(n+1)) - 1) = 2^n + n. [Numbers at the right edge.]

Prog

(Scheme, with memoization-macro definec)
(definec (A267112 n) (cond ((< n 2) n) ((even? n) (A087686 (+ 1 (A267112 (/ n 2))))) (else (A088359 (A267112 (/ (- n 1) 2))))))

Crossrefs

Inverse: A267111.

Cf. A000079, A000225, A004001, A006127, A087686, A088359, A267110.

Similar or related permutations: A003188, A054429, A276442, A233276, A233278, A276344, A276346, A276446.

Cf. also permutations A266411, A266412 and arrays A265901, A265903.

Sequence in context: A161919 A269391 A095903 * A269392 A166277 A344009

Adjacent sequences: A267109 A267110 A267111 * A267113 A267114 A267115

Keyword

nonn,tabf,look

Author

Antti Karttunen, Jan 10 2016

Status

approved