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A080542
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In binary representation: keep the first digit and rotate right the others.
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12
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1, 2, 3, 4, 6, 5, 7, 8, 12, 9, 13, 10, 14, 11, 15, 16, 24, 17, 25, 18, 26, 19, 27, 20, 28, 21, 29, 22, 30, 23, 31, 32, 48, 33, 49, 34, 50, 35, 51, 36, 52, 37, 53, 38, 54, 39, 55, 40, 56, 41, 57, 42, 58, 43, 59, 44, 60, 45, 61, 46, 62, 47, 63, 64, 96, 65, 97, 66, 98, 67, 99, 68
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OFFSET
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1,2
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COMMENTS
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Permutation of natural numbers with inverse = A080541: A080541(a(n)) = a(A080541(n)) = n;
let r(n,0)=n, r(n,k)=a(r(n,k-1)) for k>0, then r(n,floor(log_2(n))) = n and for n>1: r(n,floor(log_2(n))-1) = A080541(n).
Discarding their most significant bit, binary representations of numbers present in each cycle of this permutation form a distinct equivalence class of binary necklaces, thus there are A000031(n) separate cycles in each range [2^n .. (2^(n+1))-1] (for n >= 0) of this permutation. A256999 gives the largest number present in n's cycle. - Antti Karttunen, May 16 2015
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to necklaces
Index entries for sequences that are permutations of the natural numbers
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FORMULA
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a(n) = 2^log2(n) + floor((n-2^log2(n))/2) + (n mod 2)*2^(log2(n)-1), where log2(n) is the integer part of base-2 logarithm.
From Antti Karttunen, May 16 2015: (Start)
a(1) = 1; for n > 1, a(n) = A053644(n) + (A000035(n)*A072376(n)) + A004526(A053645(n)). [Essentially the same formula but represented with A-numbers.]
Other identities. For all n >= 1:
a(n) = A059893(A080541(A059893(n))).
a(n) = A054429(a(A054429(n))).
(End)
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EXAMPLE
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a(20) = a('10100') = '10010' = 18.
a(25) = a('11001') = '11100' = 28.
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MATHEMATICA
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kfd[n_]:=Module[{a, b}, {a, b}=TakeDrop[IntegerDigits[n, 2], 1]; FromDigits[ Join[a, RotateRight[b]], 2]]; Array[kfd, 80] (* The program uses the TakeDrop function from Mathematica version 10 *) (* Harvey P. Dale, Feb 12 2016 *)
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PROG
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(Scheme) (define (A080542 n) (if (< n 2) n (+ (A053644 n) (+ (* (A000035 n) (A072376 n)) (A004526 (A053645 n)))))) ;; Antti Karttunen, May 16 2015
(R)
nmax <- 31 # by choice
a <- 1:3
for(n in 1:nmax) for(k in 0:3)
a[4*n + k] = 2*a[2*n + (k == 1 | k == 3)] + (k == 2 | k == 3)
a
# Yosu Yurramendi, Sep 05 2020
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CROSSREFS
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Inverse: A080541.
Cf. A000031, A000035, A000523, A004526, A007088, A053644, A053645, A054429, A072376, A080414, A080544, A256999, A257250.
The set of permutations {A059893, A080541, A080542} generates an infinite dihedral group.
Sequence in context: A257683 A343152 A267105 * A330090 A072758 A104464
Adjacent sequences: A080539 A080540 A080541 * A080543 A080544 A080545
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KEYWORD
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nonn,base
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AUTHOR
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Reinhard Zumkeller, Feb 20 2003
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STATUS
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approved
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