login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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: A267676 A250299 A193497 * A267704 A191188 A285592

Adjacent sequences:  A260387 A260388 A260389 * A260391 A260392 A260393

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 31 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 23 19:46 EDT 2017. Contains 286926 sequences.