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A260446 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (0,1,0) and midword sequence (a(n)); see Comments. 3

%I

%S 0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,1,

%T 0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0,

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

%N Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (0,1,0) 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="/A260446/b260446.txt">Table of n, a(n) for n = 1..10000</a>

%F Formula: a(n) = 1 - A259599(n).

%e w(1) = 010, the initial word.

%e w(2) = 0100010 ( = 010+0+010, where + = concatenation)

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

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

%t u[1] = { 0, 1, 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]]}; u[6]

%Y Cf. A260390, A259599.

%K nonn,easy

%O 1

%A _Clark Kimberling_, Aug 22 2015

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Last modified October 23 08:57 EDT 2019. Contains 328345 sequences. (Running on oeis4.)