A230127 Number of binary strings of length n avoiding "squares" (that is, repeated blocks of the form xx) with |x| > 1.

1, 2, 4, 8, 12, 20, 26, 38, 42, 52, 56, 56, 48, 42, 32, 22, 10, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,2

Comments

Entringer et al. showed that a(n) = 0 for all n >= 19.

Links

Table of n, a(n) for n=0..81.

Nathaniel Johnston, C code for computing this sequence

R. C. Entringer, D. E. Jackson and J. A. Schatz, On nonrepetitive sequences, J. Combin. Theory Ser. A. 16 (1974), 159-164.

Example

a(4) = 12 because there are 16 binary strings of length 4, but 4 of these strings (namely 0000, 0101, 1010, and 1111) repeat a substring of length 2. Thus a(4) = 16 - 4 = 12.
a(18) = 2 because there are 2 strings of length 18 not containing any "squares" of length greater than 1: 010011000111001101 and 101100111000110010.

Mathematica

a[n_] := a[n] = Select[PadLeft[#, n]& /@ IntegerDigits[Range[0, 2^n-1], 2], {} == SequencePosition[#, {b__, b__} /; Length[{b}]>1, 1]&] // Length;
Table[Print[n, " ", a[n]]; a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 10 2021 *)

Crossrefs

Cf. A022087, A229614, A230177.

Sequence in context: A344932 A136034 A243112 * A059793 A118029 A049322

Adjacent sequences: A230124 A230125 A230126 * A230128 A230129 A230130

Keyword

nonn,fini,full,changed

Author

Nathaniel Johnston, Oct 10 2013

Status

approved