

A262313


Decimal expansion of the limit of the probability that a random binary word is an instance of the Zimin pattern "abacaba" as word length approaches infinity.


3



1, 1, 9, 4, 4, 3, 6, 9, 5, 2, 5, 2, 8, 6, 3, 3, 7, 3, 0, 0, 0, 1, 1, 8, 5, 8, 6, 1, 2, 6, 8, 8, 5, 1, 0, 4, 8, 1, 5, 9, 0, 7, 9, 8, 8, 8, 1, 6, 8, 0, 8, 3, 3, 0, 8, 6, 3, 0, 6, 5, 2, 2, 2, 0, 2, 8, 9, 1, 4, 4, 5, 5, 9, 4, 2, 1, 0, 7, 7, 6, 1, 0, 7, 2
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OFFSET

0,3


COMMENTS

Word W over alphabet L is an instance of "abacaba" provided there exists a nonerasing monoid homomorphism f:{a,b,c}*>L* such that f(W)=abacaba. For example "01011010001011010" is an instance of "abacaba" via the homomorphism defined by f(a)=010, f(b)=11, f(c)=0. For a proof of the formula or more information on Zimin words, see Rorabaugh (2015).


LINKS

Danny Rorabaugh, Table of n, a(n) for n = 0..185
D. Rorabaugh, Toward the Combinatorial Limit Theory of Free Words, University of South Carolina, ProQuest Dissertations Publishing (2015). See section 5.2.
A. I. Zimin, Blokirujushhie mnozhestva termov (Russian), Mat. Sbornik, 119 (1982), 363375; Blocking sets of terms (English), Math. USSRSbornik, 47 (1984), 353364.


FORMULA

The constant is Sum_{n=1..infinity} A003000(n)*(Sum_{i=0..infinity} G_n(i)+H_n(i)), with:
G_n(i) = (1)^i * r_n((1/2)^(2*2^i)) * (Product_{j=0..i1} s_n((1/2)^(2*2^j))) / (Product_{k=0..i} 12*(1/2)^(2*2^k)),
r_n(x) = 2*x^(2n+1)  x^(4n) + x^(5n)  2*x^(5n+1) + x^(6n),
s_n(x) = 1  2*x^(1n) + x^(n);
H_n(i) = (1)^i * u_n((1/2)^(2*2^i)) * (Product_{j=0..i1} v_n((1/2)^(2*2^j))) / (Product_{k=0..i} 12*(1/2)^(2*2^k)),
u_n(x) = 2*x^(4n+1)  x^(5n) + 2*x^(5n+1) + x^(6n),
v_n(x) = 1  2*x^(1n) + x^(n)  2*x^(12n) + x^(2n).
The inside sum is an alternating series and the outside sum has positive terms and a simple tail bound. Consequentially, we have the following bounds with any positive integers N and K:
Lower bound, Sum_{n=1..N} A003000(n)*(Sum_{i=0..2K1} G_n(i)+H_n(i));
Upper bound, (1/2)^N + Sum_{n=1..N} A003000(n)*(Sum_{i=0..2K} G_n(i)+H_n(i)).


EXAMPLE

The constant is 0.11944369525286337300011858612688510481590798881680833086306522202891445594210776107239...


CROSSREFS

Cf. A003000, A123121, A262312 (aba).
Sequence in context: A117018 A193712 A249600 * A097878 A173571 A275915
Adjacent sequences: A262310 A262311 A262312 * A262314 A262315 A262316


KEYWORD

nonn,cons


AUTHOR

Danny Rorabaugh, Sep 17 2015


STATUS

approved



