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

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

Author

Philippe Deléham, Feb 24 2004

Keywords

• easy
• nonn

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

Crossrefs

Cf. A036554, A065037, A079523, A081706.

Programs

Mathematica

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

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

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

Extensions

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