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A280218
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Number of binary necklaces of length n with no subsequence 0000.
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7
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1, 2, 3, 5, 6, 11, 15, 27, 43, 75, 125, 228, 391, 707, 1262, 2285, 4119, 7525, 13691, 25111, 46033, 84740, 156123, 288529, 533670, 989305, 1835983, 3412885, 6351031, 11834623, 22074821, 41222028, 77048131, 144148859, 269913278, 505826391, 948652695, 1780473001, 3343960175, 6284560319, 11818395345
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OFFSET
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1,2
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COMMENTS
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a(n) is the number of cyclic sequences of length n consisting of zeros and ones that do not contain four consecutive zeros provided we consider as equivalent those sequences that are cyclic shifts of each other.
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LINKS
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FORMULA
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a(n) = (1/n) * Sum_{d divides n} totient(n/d) * A073817(d).
G.f.: Sum_{k>=1} (phi(k)/k) * log(1/(1-B(x^k))) where B(x) = x*(1+x+x^2+x^3).
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EXAMPLE
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a(5)=6 because we have six binary cyclic sequences of length 5 that avoid four consecutive zeros: 00011, 00101, 00111, 01101, 01111, 11111.
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MATHEMATICA
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Table[(1/n) Sum[EulerPhi[n/d] SeriesCoefficient[(4 - 3 x - 2 x^2 - x^3)/(1 - x - x^2 - x^3 - x^4), {x, 0, d}], {d, Divisors@ n}], {n, 41}] (* Michael De Vlieger, Dec 30 2016 *)
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PROG
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(PARI) N=44; x='x+O('x^N);
B(x)=x*(1+x+x^2+x^3);
Vec(sum(k=1, N, eulerphi(k)/k * log(1/(1-B(x^k))))) \\ Joerg Arndt, Dec 29 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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