A065037 -id:A065037 - cp's OEIS frontend

A036552 List of pairs (m,2m) where m is the least unused positive number.

1, 2, 3, 6, 4, 8, 5, 10, 7, 14, 9, 18, 11, 22, 12, 24, 13, 26, 15, 30, 16, 32, 17, 34, 19, 38, 20, 40, 21, 42, 23, 46, 25, 50, 27, 54, 28, 56, 29, 58, 31, 62, 33, 66, 35, 70, 36, 72, 37, 74, 39, 78, 41, 82, 43, 86, 44, 88, 45, 90, 47, 94, 48, 96, 49, 98, 51, 102

Offset: 1

Tom Verhoeff

A permutation of the natural numbers. Inverse permutation is A065037.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
P. Erdős, R. Freud, and N. Hegyvári, Arithmetical properties of permutations of integers, Acta Mathematica Hungarica 41:1-2 (1983), pp. 169-176.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625[math.CO], 2020-2021.
Index entries for sequences that are permutations of the natural numbers

Alternating merge of A003159 and A036554. Cf. A064736, A065037.

import Data.List (delete)
a036552 n = a036552_list !! (n-1)
a036552_list = g [1..] where
   g (x:xs) = x : (2*x) : (g $ delete (2*x) xs)
-- Reinhard Zumkeller, Feb 07 2011

w = {}; Do[ w = If[ FreeQ[w, k], w = Join[w, {k, 2k}], w], {k, 100}]; w
(* Jean-François Alcover, Nov 04 2011, after Wouter Meeussen *)

apairs(N) = my(m=0, r=List(), i=0); while(#rRuud H.G. van Tol, May 09 2024

A091297 A fixed point of the morphism 0 -> 02, 1 -> 02, 2 -> 11, starting from 0.

Original entry on oeis.org

0, 2, 1, 1, 0, 2, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 0, 2, 0, 2, 1, 1, 0, 2, 0, 2, 0, 2, 1, 1, 0, 2, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 0, 2, 0, 2, 1, 1, 0, 2, 0, 2, 0, 2, 1, 1, 0, 2, 0, 2, 0

Offset: 1

Philippe Deléham, Feb 24 2004

To construct the sequence: start from the Feigenbaum sequence A035263 = 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, ..., then change 1 -> 0, 2 and 0 -> 1, 1. - Philippe Deléham, Apr 18 2004
This Feigenbaum interpretation is equivalent to writing n+1 = binary "...1 00..00 x" where x is the least significant bit and zero or more 0's.  If an odd number of 0's then a(n) = 1, otherwise a(n) = 2*x.  In a similar way, if n-1 = binary "...0 11..11 x" with an odd number of 1's then a(n)=1 and otherwise a(n) = 2*x. - Kevin Ryde, Oct 17 2020
From Mikhail Kurkov, Mar 25 2021: (Start)
This sequence can be represented as a binary tree. Each child to the right is obtained by applying mex to the parent, and each child to the left is obtained by applying mex to the set formed by the parent and its second child:
                                     ( )
                                      |
                   ...................0...................
                  2                                       1
        1......../ \........0                   2......../ \........0
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    2       0           2       1           1       0           2       1
   1 0     2 1         1 0     2 0         2 0     2 1         1 0     2 0
etc.
Here mex means smallest nonnegative missing number.
Each parent and its two children form a set {0,1,2}. (End)

Mikhail Kurkov, Table of n, a(n) for n = 1..8192
Index entries for sequences that are fixed points of mappings

Cf. A036554, A065037, A079523, A081706.

Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 2}, 1 -> {0, 2}, 2 -> {1, 1}}) ]}], {0}, 7] (* Robert G. Wilson v, Mar 03 2005 *)

a(n)={while(1, my(m=logint(n,2)); if(n==2*2^m-1, return(m%2)); if(n==2^m, return(1 + m%2)); n-=2^m)} \\ Andrew Howroyd, Oct 17 2020

a(n) = n++; my(k=valuation(n>>1,2)); if(k%2==1, 1, 2*(n%2)); \\ Kevin Ryde, Oct 17 2020

a(n) = 0 iff n = A079523(k), a(n) = 1 iff n = A081706(2*k) or n = 1 + A081706(2*k), a(n) = 2 iff n = A036554(k).
a(2*n-1) + a(2*n) = 2.
a(2*n-1) = (A065037(2*n+1) - A065037(2*n-1) - 2)/2.
From Mikhail Kurkov, Oct 10 2020: (Start)
a(2^m-1) = 1 - m mod 2, m > 0,
a(2^m) = 1 + m mod 2, m > 0,
a(2^m+k) = a(k) for 0 < k < 2^m-1, m > 1.
a(2^m-k) = 2 - a(k-1) for 1 < k <= 2^(m-1), m > 1. (End)
a(2n+1) = mex{a(n)}, a(2n) = mex{a(n),a(2n+1)} or a(2n+1) = [a(n)=0], a(2n) = 2 - [a(n)=2] for n > 0 with a(1) = 0. - Mikhail Kurkov, Mar 25 2021

More terms from Robert G. Wilson v, Mar 03 2005

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A036552 List of pairs (m,2m) where m is the least unused positive number.

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A091297 A fixed point of the morphism 0 -> 02, 1 -> 02, 2 -> 11, starting from 0.

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