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A259599 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (1,0,1) and midword sequence (a(n)); see Comments. 4

%I

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

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

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

%N Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (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.

%C a(n) = A035263(n) for n=1..31, but not n=32.

%H Clark Kimberling, <a href="/A259599/b259599.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 1 - A260446(n).

%e w(1) = 101, the initial word;

%e w(2) = 1011101 ( = 101+1+101, where + = concatenation);

%e w(3) = w(2)+0+w(2)* = 101110101011101;

%e w(4) = w(3)+1+w(3)*.

%t u[1] = {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][[k]]}; u[6]

%Y Cf. A260390, A260446.

%K nonn,easy

%O 1

%A _Clark Kimberling_, Aug 13 2015

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Last modified July 15 16:09 EDT 2019. Contains 325049 sequences. (Running on oeis4.)