A343985 a(n) = A343984(n)/2.

0, 1, 1, 4, 4, 6, 9, 10, 10, 12, 14, 16, 19, 20, 21, 22, 22, 24, 26, 28, 30, 32, 34, 36, 39, 40, 41, 42, 43, 44, 45, 46, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 94, 96

Offset

1,4

Comments

a(n) is the number of n-digit singular subwords of the Thue-Morse word A010060 that end with 0; a subword w is singular if exactly one of the words w0 and w1 is also a subword.

Links

Table of n, a(n) for n=1..66.

Example

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) = 1.

Mathematica

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 *)
u/2  (* A343985 *)

Crossrefs

Cf. A010060, A343984.

Sequence in context: A201401 A163925 A200609 * A128037 A102414 A127799

Adjacent sequences: A343982 A343983 A343984 * A343986 A343987 A343988

Keyword

nonn

Author

Clark Kimberling, May 07 2021

Status

approved