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A260455 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = 0 and midword sequence (1,null,1,null,1,null,...); see Comments. 5
0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 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. See A260390 for examples.

LINKS

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

FORMULA

a(n) = 1 - A260456(n).

EXAMPLE

w(1) = 0, the initial word.

w(2) = 010 ( = 0+2+0, where + = concatenation)

w(3) = 010010 = w(2)+null+w(2)*, where null - the empty word

w(4) = w(3)+1+w(3)*

MATHEMATICA

u[1] = {0}; m[1] = {1}; u[n_] := u[n] = Join[u[n - 1], m[n - 1], Reverse[u[n - 1]]];

m[k_] := If[OddQ[k], {1}, {}]  (* midword seq:  1, null, 1, null, 1, null, ... *)

v = u[8]  (* A240455 *)

Flatten[Position[v, 0]]   (* A260479 *)

Flatten[Position[v, 1]]   (* A260480 *)

CROSSREFS

Cf. A260390, A260479. A260480.

Sequence in context: A188014 A288600 A284468 * A189572 A287028 A327202

Adjacent sequences:  A260452 A260453 A260454 * A260456 A260457 A260458

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Aug 29 2015

STATUS

approved

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Last modified November 14 20:11 EST 2019. Contains 329129 sequences. (Running on oeis4.)