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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

%I #17 Aug 27 2015 11:20:46

%S 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,

%T 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,

%U 3,2,1,2,1,2,3,1,3,2,1,3,1,2,3,1,3,2

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

%C 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*.

%C 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.

%C As a symmetrical triangle:

%C ...............................1

%C ............................1231321

%C ........................123132121231321

%C ................1231321212313213123132121231321

%C 123132121231321312313212123132111231321212313213123132121231321

%C ...

%H Clark Kimberling, <a href="/A260449/b260449.txt">Table of n, a(n) for n = 1..10000</a>

%e w(1) = 123, the initial word.

%e w(2) = 1231321 ( = 123+1+321, where + = concatenation)

%e w(3) = w(2)+2+w(2)*

%e w(4) = w(3)+3+w(3)*

%t u[1] = {1, 2, 3}; m[1] = {u[1][[1]]};

%t u[n_] := u[n] = Join[u[n - 1], m[n - 1], Reverse[u[n - 1]]]

%t m[k_] := {u[k][[k]]}; v = u[8]; (* A260449 *)

%t Flatten[Position[v, 1]] (* A260395 *)

%t Flatten[Position[v, 2]] (* A260400 *)

%t Flatten[Position[v, 3]] (* A260398 *)

%Y Cf. A260390.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Aug 22 2015