A260445 - OEIS

%I #5 Aug 22 2015 16:38:35

%S 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,

%T 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,

%U 1,0,0,0,0,0,1,0,1,0,0,1,0,0,1,0,1,0

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

%C 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*.

%C 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.

%H Clark Kimberling, <a href="/A260445/b260445.txt">Table of n, a(n) for n = 1..10000</a>

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

%e w(1) = 001, the initial word.

%e w(2) = 0010100 ( = 001+0+100, where + = concatenation)

%e w(3) = w(2)+1+w(2)*

%e w(4) = w(3)+0+w(3)*

%t u[1] = { 0, 0, 1}; m[1] = {u[1][[1]]};

%t u[n_] := u[n] = Join[u[n - 1], m[n - 1], Reverse[u[n - 1]]];

%t m[k_] := {u[k][[k]]}; u[6]

%Y Cf. A260390, A260397.

%K nonn,easy

%O 1

%A _Clark Kimberling_, Aug 22 2015