A006156 Number of ternary squarefree words of length n.

1, 3, 6, 12, 18, 30, 42, 60, 78, 108, 144, 204, 264, 342, 456, 618, 798, 1044, 1392, 1830, 2388, 3180, 4146, 5418, 7032, 9198, 11892, 15486, 20220, 26424, 34422, 44862, 58446, 76122, 99276, 129516, 168546, 219516, 285750, 372204, 484446, 630666, 821154

Offset: 0

Author

N. J. A. Sloane, Jeffrey Shallit, Doron Zeilberger

Keywords

• nonn
• nice

Comments

a(n), n > 0, is a multiple of 3 by symmetry. - Michael S. Branicky, Jul 21 2021

Examples

			Let the alphabet be {a,b,c}. Then:
a(1)=3: a, b, c.
a(2)=6: all xy except aa, bb, cc.
a(3)=12: aba, abc, aca, acb and similar words beginning with b and c, for a total of 12.
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Hans Havermann, Table of n, a(n) for n = 0..110 [Terms 1-90 come from The Entropy of Square-Free Words by Baake, Elser, & Grimm (pages 10, 11). Terms 91-110 come from Grimm's Improved Bounds on the Number of Ternary Square-Free Words (page 3).]
• M. Baake, V. Elser and U. Grimm, The entropy of square-free words, arXiv:math-ph/9809010, 1998.
• Jean Berstel, Some recent results on squarefree words, STACS 84, Symposium of Theoretical Aspects of Computer Science Paris, 11-13, 1984, pp 14-25.
• F.-J. Brandenburg, Uniformly growing k-th power-free homomorphisms, Theoretical Computer Sci., 23 (1983), 69-82.
• J. Brinkhuis, Non-repetitive sequences on three symbols, Quart. J. Math. Oxford, 34 (1983), 145-149.
• S. Ekhad and D. Zeilberger, There are more than 2^(n/17) n-letter ternary square-free words, J. Integer Sequences, Vol. 1 (1998), Article 98.1.9.
• U. Grimm, Improved bounds on the number of ternary square-free words, J. Integer Sequences, Vol. 4 (2001), Article 01.2.7.
• Mari Huova, Combinatorics on Words. New Aspects on Avoidability, Defect Effect, Equations and Palindromes, Turku Centre for Computer Science, TUCS Dissertations No 172, April 2014.
• Mari Huova and Juhani Karhumäki, On Unavoidability of k-abelian Squares in Pure Morphic Words, Journal of Integer Sequences, Vol. 16 (2013), #13.2.9.
• R. Kolpakov, Efficient Lower Bounds on the Number of Repetition-free Words, J. Integer Sequences, Vol. 10 (2007), Article 07.3.2.
• Vladislav Makarov, Counting ternary square-free words quickly, arXiv:2012.03926 [cs.FL], 2020.
• J. Noonan and D. Zeilberger, The Goulden-Jackson cluster method: extensions, applications and implementations, arXiv:math/9806036 [math.CO], 1998.
• C. Richard and U. Grimm, On the entropy and letter frequencies of ternary square-free words, arXiv:math/0302302 [math.CO], 2003.
• A. M. Shur, Growth properties of power-free languages, Computer Science Review, Vol. 6 (2012), 187-208.
• A. M. Shur, Numerical values of the growth rates of power-free languages, arXiv:1009.4415 [cs.FL], 2010.
• Yuriy Tarannikov, The minimal density of a letter in an infinite ternary square-free word is 0.2746..., Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.2.
• Eric Weisstein's World of Mathematics, Squarefree Word

Crossrefs

Cf. A060688, A282212.

Second column of A215075, multiplied by 3!=6.

Programs

Mathematica

(* A simple solution (though not at all efficient beyond n = 12) : *) a[0] = 1; a[n_] := a[n] = Length @ DeleteCases[Tuples[Range[3], n] , {a___, b__, b__, c___} ]; s = {}; Do[Print["a[", n, "] = ", a[n]]; AppendTo[s, a[n]], {n, 0, 12}]; s (* Jean-François Alcover, May 02 2011 *)
Length/@NestList[DeleteCases[Flatten[Outer[Append, #, Range@3, 1], 1], {_, x__, x__, _}] &, {{}}, 20] (* Vladimir Reshetnikov, May 16 2016 *)

Python

def isf(s): # incrementally squarefree (check factors ending in last letter)
    for l in range(1, 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 = 3*len(sfs)
        sfsnew = set(s+i for s in sfs for i in "012" if isf(s+i))
        alst, sfs = alst+[an], sfsnew
        if verbose: print(n, an)
    return alst
print(aupton(40)) # Michael S. Branicky, Jul 21 2021

Formula

a(n) >= 2^(n/17), see Zeilberger. Let L = lim_{n->infinity} a(n)^(1/n); then L exists and Grimm proves 1.109999 < L < 1.317278. - Charles R Greathouse IV, Nov 29 2013

L exists since a(n) is submultiplicative; 1.3017597 < L < 1.3017619 (Shur 2012); the gap between the bounds can be made less than any given constant. - Arseny Shur, Apr 22 2015

Extensions

Links corrected by Eric Rowland, Sep 16 2010