m0 = 0, >>> MORSE-THUE Sequence <<< m1 = m0 & Complement(m0) = 01, m2 = m1 & Complement(m1) = 0110, . . . m = 01101001100101101001011001101001... the Morse-Thue sequence m[k] = ( number of ones in the binary representation of k ) mod 2 _______________________________________________________________________________ a(n) = #factors of the Morse sequence of length n beginning with 0. n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a(n) 1 2 3 5 6 8 10 11 12 14 16 18 20 21 22 23 24 26 28 30 Delta a(n) 1 1 2 1 2 2 1 1 2 2 2 2 1 1 1 1 2 2 2 Delta a(n) contains only 1 and 2. The sequence bears the following pattern: 1 1;2 1; 2 2 1 1; 4times 2, 4times 1; 8times 2, 8times 1; 16times 2, 16times 1; The 126 factors (substrings) of the Morse-sequence of length 80 beginning with 0. The complement strings are factors too. 00101100110100101101001100101100110100110010110100101100110100101101001100101101 00101100110100101101001100101100110100110010110100101100110100110010110011010010 00101100110100101101001100101101001011001101001011010011001011001101001100101101 00101100110100101101001100101101001011001101001100101100110100101101001100101100 00101100110100101101001100101101001011001101001100101100110100101101001100101101 00101100110100110010110011010010110100110010110011010011001011010010110011010010 00101100110100110010110011010010110100110010110011010011001011010010110011010011 00101100110100110010110011010010110100110010110100101100110100101101001100101100 00101100110100110010110100101100110100101101001100101101001011001101001100101100 00101100110100110010110100101100110100110010110011010010110100110010110011010011 00101100110100110010110100101100110100110010110011010010110100110010110100101100 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01101001100101100110100101101001100101101001011001101001011010011001011001101001 01101001100101100110100110010110100101100110100101101001100101101001011001101001 01101001100101100110100110010110100101100110100110010110011010010110100110010110 01101001100101101001011001101001011010011001011001101001100101101001011001101001 01101001100101101001011001101001011010011001011010010110011010011001011001101001 01101001100101101001011001101001100101100110100101101001100101100110100110010110 01101001100101101001011001101001100101100110100101101001100101101001011001101001 -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/