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.

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

Offset: 0

Author

Danny Rorabaugh, Sep 17 2015

Keywords

• nonn
• cons

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).

Examples

			The constant is 0.11944369525286337300011858612688510481590798881680833086306522202891445594210776107239...
		

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), 363-375; Blocking sets of terms (English), Math. USSR-Sbornik, 47 (1984), 353-364.

Crossrefs

Cf. A003000, A123121, A262312 (aba).

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..i-1} s_n((1/2)^(2*2^j))) / (Product_{k=0..i} 1-2*(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^(1-n) + x^(-n);

H_n(i) = (-1)^i * u_n((1/2)^(2*2^i)) * (Product_{j=0..i-1} v_n((1/2)^(2*2^j))) / (Product_{k=0..i} 1-2*(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^(1-n) + x^(-n) - 2*x^(1-2n) + 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..2K-1} 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)).