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A267112
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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.
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13
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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
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OFFSET
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1,2
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COMMENTS
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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:
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...................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.
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LINKS
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FORMULA
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a(1) = 1; after which, a(2n) = A087686(1+a(n)), a(2n+1) = A088359(a(n)).
As a composition of other permutations:
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.]
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PROG
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(Scheme, with memoization-macro definec)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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