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A262313
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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.
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3
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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
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0,3
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COMMENTS
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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).
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LINKS
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FORMULA
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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)).
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EXAMPLE
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The constant is 0.11944369525286337300011858612688510481590798881680833086306522202891445594210776107239...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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