function [ n, thue ] = thue_ternary_next ( n, thue )
%*****************************************************************************80
%
%% thue_ternary_next() returns the next element in a ternary Thue sequence.
%
% Discussion:
%
% Thue was interested in showing that there were arbitrarily long
% sequences of digits which never displayed a pair of contiguous
% repetitions of any length. That is, there was no occurrence of
% "00" or "1010" or "121121", anywhere in the string. This makes
% the string "squarefree".
%
% To do this, he demonstrated a way to start with a single digit,
% and to repeatedly apply a series of transformation rules to each
% digit of the sequence, deriving nonrepeating sequences of ever
% greater length.
%
% In this example, the digits allowed are ternary, that is, just
% "0", "1" and "2". The replacement rules are:
%
% "0" -> "12"
% "1" -> "102"
% "2" -> "0"
%
% This routine produces the next Thue sequence in a given series.
% However, the input sequence must be a Thue sequence in order for
% us to guarantee that the output sequence will also have the
% nonrepetition property.
%
% Also, enough space must be set aside in THUE to hold the
% output sequence. This will never be more than 3 times the input
% value of N.
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 20 June 2004
%
% Author:
%
% John Burkardt
%
% Reference:
%
% Brian Hayes,
% Third Base,
% American Scientist,
% Volume 89, Number 6, pages 490-494, November-December 2001.
%
% Input:
%
% integer N, the length of the input sequence.
%
% integer THUE(N), the initial Thue sequence.
%
% Output:
%
% integer N, the length of the output sequence.
%
% integer THUE(N), the result of applying the substitution rules once.
%
n2 = 0;
thue2 = [];
for i = 1 : n
if ( thue(i) == 0 )
n2 = n2 + 1;
thue2(n2) = 1;
n2 = n2 + 1;
thue2(n2) = 2;
elseif ( thue(i) == 1 )
n2 = n2 + 1;
thue2(n2) = 1;
n2 = n2 + 1;
thue2(n2) = 0;
n2 = n2 + 1;
thue2(n2) = 2;
elseif ( thue(i) == 2 )
n2 = n2 + 1;
thue2(n2) = 0;
else
fprintf ( 1, '\n' );
fprintf ( 1, 'THUE_TERNARY_NEXT - Fatal error!\n' );
fprintf ( 1, ' The input sequence contains a non-ternary digit\n' );
fprintf ( 1, ' THUE(%d) = %d\n', i, thue(i) );
error ( 'THUE_TERNARY_NEXT - Fatal error!\n' );
end
end
n = n2;
thue = thue2;
return
end