A260455 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = 0 and midword sequence (1,null,1,null,1,null,...); see Comments.

0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0

Offset

1

Comments

Below, w* denotes the reversal of a word w, and "sequence" and "word" are interchangable. An infinite word is palindromic if it has infinitely many initial subwords w such that w = w*.

Many infinite palindromic words (a(1),a(2),...) are determined by an initial word w and a midword sequence (m(1),m(2),...) of palindromes, as follows: for given w of length k, take w(1) = w = (a(1),a(2),...,a(k)). Form the palindrome w(2) = w(1)m(1)w(1)* by concatenating w(1), m(1), and w(1)*. Continue inductively; i.e., w(n+1) = w(n)m(n)w(n)* for all n >= 1. See A260390 for examples.

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Formula

a(n) = 1 - A260456(n).

Example

w(1) = 0, the initial word.
w(2) = 010 ( = 0+2+0, where + = concatenation)
w(3) = 010010 = w(2)+null+w(2)*, where null - the empty word
w(4) = w(3)+1+w(3)*

Mathematica

u[1] = {0}; m[1] = {1}; u[n_] := u[n] = Join[u[n - 1], m[n - 1], Reverse[u[n - 1]]];
m[k_] := If[OddQ[k], {1}, {}]  (* midword seq:  1, null, 1, null, 1, null, ... *)
v = u[8]  (* A240455 *)
Flatten[Position[v, 0]]   (* A260479 *)
Flatten[Position[v, 1]]   (* A260480 *)

Crossrefs

Cf. A260390, A260479. A260480.

Sequence in context: A188014 A288600 A284468 * A189572 A287028 A327202

Adjacent sequences: A260452 A260453 A260454 * A260456 A260457 A260458

Keyword

nonn,easy

Author

Clark Kimberling, Aug 29 2015

Status

approved