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A091297 A fixed point of the morphism 0 -> 02, 1 -> 02, 2 -> 11, starting from 0. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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)

LINKS

Mikhail Kurkov, Table of n, a(n) for n = 1..8192

Index entries for sequences that are fixed points of mappings

FORMULA

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

MATHEMATICA

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

PROG

(PARI) 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

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

CROSSREFS

Cf. A036554, A065037, A079523, A081706.

Sequence in context: A054009 A339210 A176451 * A166712 A035183 A178101

Adjacent sequences:  A091294 A091295 A091296 * A091298 A091299 A091300

KEYWORD

easy,nonn

AUTHOR

Philippe Deléham, Feb 24 2004

EXTENSIONS

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

STATUS

approved

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Last modified May 7 20:36 EDT 2021. Contains 343652 sequences. (Running on oeis4.)