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Search: id:A154436
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| A154436 |
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Permutation of non-negative integers induced by Lamplighter group generating wreath recursion, variant 1: a = s(a,b), b = (a,b), starting from the state a. |
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+0
14
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| 0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13, 9, 8, 10, 11, 31, 30, 28, 29, 25, 24, 26, 27, 19, 18, 16, 17, 21, 20, 22, 23, 63, 62, 60, 61, 57, 56, 58, 59, 51, 50, 48, 49, 53, 52, 54, 55, 39, 38, 36, 37, 33, 32, 34, 35, 43, 42, 40, 41, 45, 44, 46, 47, 127, 126, 124, 125, 121, 120
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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This permutation is induced by the first Lamplighter group generating wreath recursion a = s(a,b), b = (a,b) (i.e. binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on page 104 of Bondarenko, Grigorchuk, et al. paper, starting from the active (swapping) state a and rewriting bits from the second most significant bit to the least significant end. It is the same automaton as given in figure 1 on page 211 of Grigorchuk and Zuk paper. Note that the fourth wreath recursion on page 104 of Bondarenko, et al. paper induces similarly the binary reflected Gray code A003188 (A054429-reflected conjugate of this permutation) and the second one induces Gray Code's inverse permutation A006068.
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LINKS
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A. Karttunen, Table of n, a(n) for n = 0..2047
Index entries for sequences that are permutations of the natural numbers
S. Wolfram, R. Lamy, Discussion on the NKS Forum
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EXAMPLE
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312 = 100111000 in binary. Starting from the second most significant bit and, as we begin with the swapping state a, we complement the bits up to and including the first one encountered and so the beginning of the binary expansion is complemented as 1110....., then, as we switch to the inactive state b, the following bits are kept same, up to and including the first zero encountered, after which the binary expansion is 1110110.., after which we switch again to the complementing mode (state a) and we obtain 111011011, which is 475's binary representation, thus a(312)=475.
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PROGRAM
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(MIT Scheme:) (define (A154436 n) (if (< n 2) n (let loop ((maskbit (A072376 n)) (state 1) (z 1)) (if (zero? maskbit) z (let ((dombit (modulo (floor->exact (/ n maskbit)) 2))) (cond ((= state dombit) (loop (floor->exact (/ maskbit 2)) (- 1 state) (+ z z (modulo (- state dombit) 2)))) (else (loop (floor->exact (/ maskbit 2)) state (+ z z (modulo (- state dombit) 2))))))))))
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CROSSREFS
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Inverse: A154435. a(n) = A059893(A154438(A059893(n))) = A054429(A003188(A054429(n))). Corresponds to A122302 in the group of Catalan bijections. Cf. also A153141-A153142, A154439-A154448, A072376.
Sequence in context: A160679 A153141 A006068 this_sequence A072764 A130328 A083569
Adjacent sequences: A154433 A154434 A154435 this_sequence A154437 A154438 A154439
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KEYWORD
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nonn,base
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AUTHOR
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Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 17 2009
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