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 A260390 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (1,0) and midword sequence (a(n)); see Comments. 27
 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0 (list; graph; refs; listen; history; text; internal format)
 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. Examples follow: initial word    midword sequence   inf. palindr. word   |w(n)| w(1) = 10         m(i) = a(i)         A260390          A083329 w(1) = 01         m(i) = a(i)         A260393          A083329 w(1) = 011        m(i) = a(i)         A260394          A000225 w(1) = 110        m(i) = a(i)         A260397          A000225 w(1) = 101        m(i) = a(i)         A035263          A000225 w(1) = 100        m(i) = a(i)         A260444          A000225 w(1) = 001        m(i) = a(i)         A260445          A000225 w(1) = 010        m(i) = a(i)         A260446          A000225 w(1) = 0          m(i) = i            A007814          A000225 w(1) = 123        m(i) = a(i)         A260449          A000225 w(1) = 132        m(i) = a(i)         A260450          A000225 w(1) = 231        m(i) = a(i)         A260451          A000225 w(1) = 213        m(i) = a(i)         A260452          A000225 w(1) = 321        m(i) = a(i)         A260453          A000225 w(1) = 312        m(i) = a(i)         A260454          A000225 w(1) = 0          (see A260455)       A260455          A081254 (conjectured) w(1) = 1          (see A260456)       A260456          A081254 (conjectured) As a sort of (obvious) converse of the above method for constructing infinite palindromic words, every such word is determined by an initial segment w(1) and a midword sequence (m(n)), where terms of the latter may be the empty word. LINKS Clark Kimberling, Table of n, a(n) for n = 1..10000 FORMULA a(n) = 1 - A260393(n). EXAMPLE w(1) = 10, the initial word. w(2) = 10101 ( = 10+1+01, where + = concatenation) w(3) = 10101010101 = w(2)+0+w(2)* w(4) = w(3)+1+w(3)* MATHEMATICA u[1] = {1, 0}; m[1] = {u[1][[1]]}; u[n_] := u[n] = Join[u[n - 1], m[n - 1], Reverse[u[n - 1]]]; Table[Length[u[n]], {n, 1, 20}]  (* A083329 *) Flatten[Position[u[8], 0]]   (* A260391 *) Flatten[Position[u[8], 1]]   (* A260392 *) CROSSREFS Cf. A083329, A260392, A260394. Sequence in context: A290079 A250299 A193497 * A287795 A267704 A191188 Adjacent sequences:  A260387 A260388 A260389 * A260391 A260392 A260393 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jul 31 2015 STATUS approved

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