A260445 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (0,0,1) and midword sequence (a(n)); see Comments.

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

Formula: a(n) = 1 - A260397(n).

Example

w(1) = 001, the initial word.
w(2) = 0010100 ( = 001+0+100, where + = concatenation)
w(3) = w(2)+1+w(2)*
w(4) = w(3)+0+w(3)*

Mathematica

u[1] = { 0, 0, 1}; m[1] = {u[1][[1]]};
u[n_] := u[n] = Join[u[n - 1], m[n - 1], Reverse[u[n - 1]]];
m[k_] := {u[k][[k]]}; u[6]

Crossrefs

Cf. A260390, A260397.

Sequence in context: A189820 A111406 A353463 * A156731 A254379 A288257

Adjacent sequences: A260442 A260443 A260444 * A260446 A260447 A260448

Keyword

nonn,easy

Author

Clark Kimberling, Aug 22 2015

Status

approved