A230177 -id:A230177 - cp's OEIS frontend

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

Nathaniel Johnston, Oct 10 2013

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

			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.

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.

Cf. A022087, A229614, A230177.

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 *)

A229614 Number of binary strings of length n avoiding "squares" (that is, repeated blocks of the form xx) with |x| > 2.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 56, 104, 178, 314, 536, 930, 1558, 2666, 4482, 7574, 12686, 21360, 35812, 60152, 100812, 169122, 283498, 475356, 796292, 1334558, 2235888, 3746534, 6276048, 10515080, 17614726, 29510362, 49434792, 82815016, 138729368, 232399846, 389306052

Offset: 0

Jeffrey Shallit, Sep 26 2013

nonn

Entringer et al. showed that this sequence is always nonzero (in contrast with A230127, which is zero for all n >= 19). - Nathaniel Johnston, Oct 10 2013

For n >= 1, terms are even by symmetry. - Michael S. Branicky, Nov 11 2021

			For n = 6 there are 8 strings omitted, namely 000000, 001001, ..., 111111, so a(6) = 64 - 8 = 56.

Michael S. Branicky, Table of n, a(n) for n = 0..42
Michael S. Branicky, Python program
R. C. Entringer, D. E. Jackson and J. A. Schatz, On nonrepetitive sequences, J. Combin. Theory Ser. A. 16 (1974), 159-164.

Cf. A230127, A230177, A230216.

a[n_] := a[n] = Select[PadLeft[#, n]& /@ IntegerDigits[Range[0, 2^n-1], 2], {} == SequencePosition[#, {b__, b__} /; Length[{b}] >= 3, 1]&] // Length;
Table[Print[n, " ", a[n]]; a[n], {n, 0, 22}] (* Jean-François Alcover, Nov 11 2021 *)

# see link for a faster program based on binary operations
def isf(s): # incrementally squarefree (check factors ending in last letter)
    for l in range(3, len(s)//2 + 1):
        if s[-2*l:-l] == s[-l:]: return False
    return True
def aupton(nn, verbose=False):
    alst, sfs = [1], set("0")
    for n in range(1, nn+1):
        an = 2*len(sfs) # by symmetry
        sfsnew = set(s+i for s in sfs for i in "01" if isf(s+i))
        alst, sfs = alst+[an], sfsnew
        if verbose: print(n, an)
    return alst
print(aupton(24)) # Michael S. Branicky, Nov 11 2021

a(23)-a(36) from Nathaniel Johnston, Oct 10 2013

A230216 Number of binary strings of length n avoiding "squares" (that is, repeated blocks of the form xx) with |x| = 3.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 56, 104, 192, 352, 648, 1192, 2192, 4032, 7416, 13640, 25088, 46144, 84872, 156104, 287120, 528096, 971320, 1786536, 3285952, 6043808, 11116296, 20446056, 37606160, 69168512, 127220728, 233995400, 430384640, 791600768, 1455980808

Offset: 0

Nathaniel Johnston, Oct 11 2013

			For n = 6 there are 8 strings omitted, namely 000000, 001001, ..., 111111, so a(6) = 64-8 = 56.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000073, A022087, A229614, A230127, A230177.

Vec((1 + x + x^2 + x^3 + 2*x^4 + 4*x^5) / (1 - x - x^2 - x^3) + O(x^40)) \\ Colin Barker, Aug 09 2019

a(n) = 8*A000073(n) for n >= 3.
From Colin Barker, Aug 09 2019: (Start)
G.f.: (1 + x + x^2 + x^3 + 2*x^4 + 4*x^5) / (1 - x - x^2 - x^3).
a(n) = a(n-1) + a(n-2) + a(n-3) for n>5.
(End)

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

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A229614 Number of binary strings of length n avoiding "squares" (that is, repeated blocks of the form xx) with |x| > 2.

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A230216 Number of binary strings of length n avoiding "squares" (that is, repeated blocks of the form xx) with |x| = 3.

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