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A343984
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a(n) = number of n-digit singular subwords of the Thue-Morse word A010060; see Comments.
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2
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0, 2, 2, 8, 8, 12, 18, 20, 20, 24, 28, 32, 38, 40, 42, 44, 44, 48, 52, 56, 60, 64, 68, 72, 78, 80, 82, 84, 86, 88, 90, 92, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 158, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178
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OFFSET
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1,2
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COMMENTS
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A subword w of A010060 is singular if exactly one of the words w0 and w1 is also a subword.
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LINKS
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EXAMPLE
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The six 3-digit subwords of A010060 are 001, 010, 011, 100, 101, 110. Each, except for 011 and 100, is the initial 3-letter word of two 4-letter subwords. Thus, a(3) = 2.
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MATHEMATICA
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TM = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 12]; (* A010060 *)
t[k_, n_] := t[k, n] = Take[TM, {n, n + k - 1}];
c[k_] := c[k] = Union[Table[t[k, n], {n, 1, Length[TM] - k + 1}]];
s[n_] := s[n] = Select[c[n], ! MemberQ[c[n + 1],
Join[#, {0}]] || !MemberQ[c[n + 1], Join[#, {1}]] &]
Table[s[n], {n, 1, 8}]
u = Table[Length[s[n]], {n, 1, 60}] (* A343984 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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