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%I #7 Nov 09 2015 16:15:55
%S 1,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,1,1,1,0,
%T 0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,
%U 0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0
%N Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (1,0,0) and midword sequence (a(n)); see A260390.
%C Below, w* denotes the reversal of a word w, and "sequence" and "word" are interchangeable. 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 a guide to related sequences.
%e w(1) = 100, the initial word.
%e w(2) = 1001001 ( = 100+1+001, where + = concatenation)
%e w(3) = w(2)+0+w(2)*
%e w(4) = w(3)+1+w(3)*
%t u[1] = {1, 0, 0}; 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]]}
%t v = u[6] (* A260444 *)
%Y Cf. A260390.
%K nonn,easy
%O 1
%A _Clark Kimberling_, Oct 31 2015
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