Fraenkel word: Difference between revisions
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where ''k'' > ''i'', and ''F''<sub>0</sub> is the empty word. Since '''k''' occurs as the middle letter of ''F''<sub>''k''+1</sub>, there is a copy of ''F''<sub>''k''</sub> that follows '''k'''; ''F''<sub>''k''</sub> has ''F''<sub>i</sub> as a prefix. Thus ''F''<sub>''n''</sub> has a subword | where ''k'' > ''i'', and ''F''<sub>0</sub> is the empty word. Since '''k''' occurs as the middle letter of ''F''<sub>''k''+1</sub>, there is a copy of ''F''<sub>''k''</sub> that follows '''k'''; ''F''<sub>''k''</sub> has ''F''<sub>i</sub> as a prefix. Thus ''F''<sub>''n''</sub> has a subword | ||
'''i'''''F''<sub>i</sub>'''k'''''F''<sub>i</sub>'''i''', | '''i'''''F''<sub>''i''</sub>'''k'''''F''<sub>''i''</sub>'''i''', | ||
as desired, since {{!}}''F''<sub>i</sub>{{!}} = 2<sup>''i''</sup> − 1. | as desired, since {{!}}''F''<sub>i</sub>{{!}} = 2<sup>''i''</sup> − 1. | ||