A260456 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = 1 and midword sequence (0,null,0,null,0,null,...); see Comments.

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

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

Jean-Paul Allouche, On the morphism 1 -> 121, 2 -> 12221, CNRS France, 2024. See pp. 2, 7.

Jean-Paul Allouche, On the morphism 1 -> 121, 2 -> 12221, Preprint, 2024 [Local copy, with permission]

Formula

a(n) = 1 - A260455(n).

Example

w(1) = 1, the initial word.
w(2) = 101 ( = 1+0+1, where + = concatenation)
w(3) = 101101 = w(2)+null+w(2)*, where null - the empty word
w(4) = w(3)+0+w(3)*

Mathematica

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

Crossrefs

Cf. A260390, A260455.

Sequence in context: A369736 A128174 A096055 * A125144 A115198 A005614

Adjacent sequences: A260453 A260454 A260455 * A260457 A260458 A260459

Keyword

nonn,easy

Author

Clark Kimberling, Aug 29 2015

Status

approved