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A260397
Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (1,1,0) and midword sequence (a(n)); see Comments.
6
1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1
OFFSET
1
COMMENTS
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.
LINKS
FORMULA
a(n) = 1 - A260445(n).
EXAMPLE
w(1) = 110, the initial word;
w(2) = w(1)+a(1)+w(1)* = 1101011 ( = 110+1+011, where + = concatenation);
w(3) = w(2)+a(2)+w(2)* = 110101111101011 (= 1101011+1+1101011);
w(4) = w(3)+a(3)+w(3)* = w(3)+0+w(3)*.
MATHEMATICA
u[1] = {1, 1, 0}; m[1] = {u[1][[1]]};
u[n_] := u[n] = Join[u[n - 1], m[n - 1], Reverse[u[n - 1]]];
m[k_] := {u[k][[k]]}; u[6]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 13 2015
STATUS
approved