A260393 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (0,1) and midword sequence (a(n)); see Comments.

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

Formula

a(n) = 1 - A260390(n).

Example

w(1) = 01, the initial word.
w(2) = 01010 ( = 01+0+10, where + = concatenation)
w(3) = 01010101010 = w(2)+1+w(2)*
w(4) = w(3)+1+w(3)*

Mathematica

u[1] = {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 - 1][[k]]}; u[8]

Crossrefs

Cf. A260390.

Sequence in context: A285685 A287773 A173923 * A360113 A361123 A125122

Adjacent sequences: A260390 A260391 A260392 * A260394 A260395 A260396

Keyword

nonn,easy

Author

Clark Kimberling, Jul 31 2015

Status

approved