%I
%S 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,
%T 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,
%U 0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1
%N Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (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="/A260393/b260393.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = 1  A260390(n).
%e w(1) = 01, the initial word.
%e w(2) = 01010 ( = 01+0+10, where + = concatenation)
%e w(3) = 01010101010 = w(2)+1+w(2)*
%e w(4) = w(3)+1+w(3)*
%t u[1] = {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  1][[k]]}; u[8]
%Y Cf. A260390.
%K nonn,easy
%O 1
%A _Clark Kimberling_, Jul 31 2015
