A260454 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (3,2,1) and midword sequence (a(n)); see Comments.

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

Offset

1,1

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

Example

w(1) = 321, the initial word.
w(2) = 3213123 ( = 321+3+123, where + = concatenation)
w(3) = w(2)+2+w(2)*
w(4) = w(3)+1+w(3)*

Mathematica

u[1] = {3, 2, 1}; 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]

Crossrefs

Cf. A260390, A260453.

Sequence in context: A112745 A205564 A036585 * A164848 A213514 A130827

Adjacent sequences: A260451 A260452 A260453 * A260455 A260456 A260457

Keyword

nonn,easy

Author

Clark Kimberling, Aug 29 2015

Status

approved