function [ n, thue ] = thue_binary_next ( n, thue )
%*****************************************************************************80
%
%% thue_binary_next() returns the next element in a binary Thue sequence.
%
% Discussion:
%
% Thue demonstrated that arbitrarily long sequences of 0's and
% 1's could be generated which had the "cubefree" property. In
% other words, for a given string S, there was no substring W
% such that S contained "WWW". In fact, a stronger result holds:
% if "a" is the first letter of W, it is never the case that S
% contains the substring "WWa".
%
% In this example, the digits allowed are binary, that is, just
% "0" and "1". The replacement rules are:
%
% "0" -> "01"
% "1" -> "10"
%
% This routine produces the next binary 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
% cubic nonrepetition property.
%
% Also, enough space must be set aside in THUE to hold the
% output sequence. This will always be twice the input
% value of N.
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 20 June 2004
%
% Author:
%
% John Burkardt
%
% 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) = 0;
n2 = n2 + 1;
thue2(n2) = 1;
elseif ( thue(i) == 1 )
n2 = n2 + 1;
thue2(n2) = 1;
n2 = n2 + 1;
thue2(n2) = 0;
else
fprintf ( 1, '\n' );
fprintf ( 1, 'THUE_BINARY_NEXT - Fatal error!\n' );
fprintf ( 1, ' The input sequence contains a non-binary digit\n' );
fprintf ( 1, ' THUE(%d) = %d\n', i, thue(i) );
error ( 'THUE_BINARY_NEXT - Fatal error!' );
end
end
n = n2;
thue = thue2;
return
end