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A260456
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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.
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3
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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, 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, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1
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OFFSET
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1
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COMMENTS
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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*.
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.
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LINKS
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FORMULA
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EXAMPLE
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w(1) = 1, the initial word.
w(2) = 101 ( = 1+0+1, where + = concatenation)
w(3) = 101101 = w(2)+null+w(2)*, where null - the empty word
w(4) = w(3)+0+w(3)*
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MATHEMATICA
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u[1] = {1}; m[1] = {0}; u[n_] := u[n] = Join[u[n - 1], m[n - 1], Reverse[u[n - 1]]];
m[k_] := If[OddQ[k], {0}, {}] (* midword seq: 1, null, 1, null, 1, null, ... *)
u[8]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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