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%I #6 Sep 07 2015 12:33:05
%S 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,
%T 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,
%U 1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1
%N 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.
%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="/A260456/b260456.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = 1 - A260455(n).
%e w(1) = 1, the initial word.
%e w(2) = 101 ( = 1+0+1, where + = concatenation)
%e w(3) = 101101 = w(2)+null+w(2)*, where null - the empty word
%e w(4) = w(3)+0+w(3)*
%t u[1] = {1}; m[1] = {0}; u[n_] := u[n] = Join[u[n - 1], m[n - 1], Reverse[u[n - 1]]];
%t m[k_] := If[OddQ[k], {0}, {}] (* midword seq: 1,null,1,null,1,null,... *)
%t u[8]
%Y Cf. A260390, A260455.
%K nonn,easy
%O 1
%A _Clark Kimberling_, Aug 29 2015
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