A267109 -id:A267109 - cp's OEIS frontend

A267111 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = 2a(n-1), a(A088359(n)) = 1+2a(n), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001, the Hofstadter-Conway $10000 sequence.

1, 2, 3, 4, 5, 7, 6, 8, 9, 11, 15, 10, 13, 14, 12, 16, 17, 19, 23, 31, 18, 21, 27, 22, 29, 30, 20, 25, 26, 28, 24, 32, 33, 35, 39, 47, 63, 34, 37, 43, 55, 38, 45, 59, 46, 61, 62, 36, 41, 51, 42, 53, 54, 44, 57, 58, 60, 40, 49, 50, 52, 56, 48, 64, 65, 67, 71, 79, 95, 127, 66, 69, 75, 87, 111, 70, 77, 91, 119, 78, 93, 123

Offset: 1

Antti Karttunen, Jan 10 2016

Antti Karttunen, Table of n, a(n) for n = 1..8192
Antti Karttunen, Entanglement Permutations, 2016-2017
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

Inverse: A267112.
Cf. A004001, A080677, A087686, A088359, A093879, A267108, A267109.
Similar or related permutations: A006068, A054429, A276441, A233275, A233277, A276343, A276345, A276445.
Cf. also permutations A266411, A266412 and arrays A265901, A265903.

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = 2*a(A080677(n)-1), otherwise [when n is in A088359], a(n) = 1 + 2*a(A004001(n)-1).
Equally, for n > 1, if A093879(n-1) = 0, a(n) = 2*a(n - A004001(n)), otherwise a(n) = 1 + 2*a(A004001(n)-1). [Above formula in a more symmetric form.]
As a composition of other permutations:
a(n) = A054429(A276441(n)).
a(n) = A233275(A276343(n)).
a(n) = A233277(A276345(n)).
a(n) = A006068(A276445(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.]

A265332 a(n) is the index of the column in A265901 where n appears; also the index of the row in A265903 where n appears.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 3, 5, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 7, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4

Offset: 1

Antti Karttunen, Jan 09 2016

nonn

If all 1's are deleted, the remaining terms are the sequence incremented. - after Franklin T. Adams-Watters Oct 05 2006 comment in A051135.

Ordinal transform of A162598.

			Illustration how the sequence can be constructed by concatenating the frequency counts Q_n of each successive level n of A004001-tree:
--
             1                                      Q_0 = (1)
             |
            _2__                                    Q_1 = (2)
           /    \
         _3    __4_____                             Q_2 = (1,3)
        /     /  |     \
      _5    _6  _7    __8___________                Q_3 = (1,1,2,4)
     /     /   / |   /  |  \        \
   _9    10  11 12  13  14  15___    16_________    Q_4 = (1,1,1,2,1,2,3,5)
  /     /   /  / |  /  / |   |\  \   | \  \  \  \
17    18  19 20 21 22 23 24 25 26 27 28 29 30 31 32
--
The above illustration copied from the page 229 of Kubo and Vakil paper (page 5 in PDF).

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.

Essentially same as A051135 apart from the initial term, which here is set as a(1)=1.
Cf. A004001, A265901, A265903.
Cf. A162598 (corresponding other index).
Cf. A265754.
Cf. also A267108, A267109, A267110.

terms = 120;
h[1] = 1; h[2] = 1;
h[n_] := h[n] = h[h[n - 1]] + h[n - h[n - 1]];
seq[nmax_] := seq[nmax] = (Length /@ Split[Sort @ Array[h, nmax, 2]])[[;; terms]];
seq[nmax = 2 terms];
seq[nmax += terms];
While[seq[nmax] != seq[nmax - terms], nmax += terms];
seq[nmax] (* Jean-François Alcover, Dec 19 2021 *)

(define (A265332 n) (if (= 1 n) 1 (A051135 n)))

a(1) = 1; for n > 1, a(n) = A051135(n).

A267110 If A051135(n) = 1, then a(n) = A004001(n) - 1, otherwise a(n) = n - A004001(n).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 5, 7, 6, 7, 8, 8, 9, 10, 11, 9, 12, 13, 10, 14, 11, 12, 15, 13, 14, 15, 16, 16, 17, 18, 19, 20, 17, 21, 22, 23, 18, 24, 25, 19, 26, 20, 21, 27, 28, 22, 29, 23, 24, 30, 25, 26, 27, 31, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 33, 38, 39, 40, 41, 34, 42, 43, 44, 35, 45, 46, 36, 47

Offset: 1

Antti Karttunen, Jan 16 2016

For n > 1, a(n) gives the contents of the parent of the node which contains n in A267112-tree.
Each n > 0 occurs exactly twice, in positions A088359(n) and A087686(n+1).
The sequence maps each n > 1 to a number which is one digit shorter in binary system (cf. "Other identities"). This follows because A004001 is monotonic and A004001(2^n) = 2^(n-1) (see properties (2) and (3) given on page 227 of Kubo & Vakil paper, or page 3 in PDF), and also how the frequency counts Q_n for A004001 are recursively constructed (see Kubo & Vakil paper, p. 229 or A265332 for the illustration).

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

Cf. A004001, A051135, A070939, A087686, A088359, A265332, A267108, A267109, A267112.

(define (A267110 n) (if (= 1 (A051135 n)) (- (A004001 n) 1) (- n (A004001 n))))

If A051135(n) = 1 [Equally: if A265332(n) = 1], then a(n) = A004001(n) - 1, otherwise a(n) = n - A004001(n).
Other identities. For all n >= 2:
A070939(a(n)) = A070939(n) - 1. [See Comments section.]

A267108 a(n) = A000120(A267111(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 2, 3, 3, 2, 1, 2, 3, 4, 5, 2, 3, 4, 3, 4, 4, 2, 3, 3, 3, 2, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 3, 4, 5, 4, 5, 5, 2, 3, 4, 3, 4, 4, 3, 4, 4, 4, 2, 3, 3, 3, 3, 2, 1, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 3, 4, 5, 6, 4, 5, 6, 5, 6, 6, 2, 3, 4, 5, 3, 4, 5, 4, 5, 5, 3, 4, 5, 4, 5, 5, 4, 5, 5, 5, 2, 3, 4, 3, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 4

Offset: 1

Antti Karttunen, Jan 16 2016

nonn

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

Cf. A000120, A004001, A070939, A088359, A265332, A267109, A267110, A267111.

a(1) = 1; for n > 1, if A265332(n) = 1 [when n is one of the terms of A088359], a(n) =  1 + a(A004001(n)-1), otherwise a(n) = a(n-A004001(n)).
a(n) = A000120(A267111(n)).
Other identities. For all n >= 1:
a(n) = A070939(n) - A267109(n).

cp's OEIS Frontend

A267111 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = 2a(n-1), a(A088359(n)) = 1+2a(n), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001, the Hofstadter-Conway $10000 sequence.

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Formula

A265332 a(n) is the index of the column in A265901 where n appears; also the index of the row in A265903 where n appears.

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Mathematica

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Formula

A267110 If A051135(n) = 1, then a(n) = A004001(n) - 1, otherwise a(n) = n - A004001(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A267108 a(n) = A000120(A267111(n)).

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Author

Keywords

Links

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Formula

cp's OEIS Frontend

A267111 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = 2*a(n-1), a(A088359(n)) = 1+2*a(n), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001, the Hofstadter-Conway $10000 sequence.

Views

Author

Keywords

Links

Crossrefs

Formula

A265332 a(n) is the index of the column in A265901 where n appears; also the index of the row in A265903 where n appears.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A267110 If A051135(n) = 1, then a(n) = A004001(n) - 1, otherwise a(n) = n - A004001(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A267108 a(n) = A000120(A267111(n)).

Views

Author

Keywords

Links

Crossrefs

Formula

A267111 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = 2a(n-1), a(A088359(n)) = 1+2a(n), where A088359 and A087686 = numbers that occur only once (resp. more than once) in A004001, the Hofstadter-Conway $10000 sequence.