A260449 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (1,2,3) and midword sequence (a(n)); see Comments.

1, 2, 3, 1, 3, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 1, 1, 2, 3, 1, 3, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2

Offset

1,2

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.

As a symmetrical triangle:

...............................1

............................1231321

........................123132121231321

................1231321212313213123132121231321

123132121231321312313212123132111231321212313213123132121231321

...

Links

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

Example

w(1) = 123, the initial word.
w(2) = 1231321 ( = 123+1+321, where + = concatenation)
w(3) = w(2)+2+w(2)*
w(4) = w(3)+3+w(3)*

Mathematica

u[1] = {1, 2, 3}; 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]]}; v = u[8];  (* A260449 *)
Flatten[Position[v, 1]]  (* A260395 *)
Flatten[Position[v, 2]]  (* A260400 *)
Flatten[Position[v, 3]]  (* A260398 *)

Crossrefs

Cf. A260390.

Sequence in context: A128222 A057039 A369010 * A135511 A007413 A277750

Adjacent sequences: A260446 A260447 A260448 * A260450 A260451 A260452

Keyword

nonn,easy

Author

Clark Kimberling, Aug 22 2015

Status

approved