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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A023678 A128222 A057039 * A135511 A007413 A277750

Adjacent sequences:  A260446 A260447 A260448 * A260450 A260451 A260452

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Aug 22 2015

STATUS

approved

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Last modified November 16 07:06 EST 2018. Contains 317258 sequences. (Running on oeis4.)