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A260456 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. 3
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 (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. See A260390 for examples.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = 1 - A260455(n).

EXAMPLE

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)*

MATHEMATICA

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]

CROSSREFS

Cf. A260390, A260455.

Sequence in context: A249866 A128174 A096055 * A125144 A115198 A005614

Adjacent sequences:  A260453 A260454 A260455 * A260457 A260458 A260459

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Aug 29 2015

STATUS

approved

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Last modified November 17 18:24 EST 2019. Contains 329241 sequences. (Running on oeis4.)