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A051841
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Number of binary Lyndon words with an even number of 1's.
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15
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1, 0, 1, 1, 3, 4, 9, 14, 28, 48, 93, 165, 315, 576, 1091, 2032, 3855, 7252, 13797, 26163, 49929, 95232, 182361, 349350, 671088, 1290240, 2485504, 4792905, 9256395, 17894588, 34636833, 67106816, 130150493, 252641280, 490853403, 954429840, 1857283155, 3616800768, 7048151355, 13743869130, 26817356775
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OFFSET
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1,5
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COMMENTS
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Also number of trace 0 irreducible polynomials over GF(2).
Also number of trace 0 Lyndon words over GF(2).
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REFERENCES
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May, Robert M. "Simple mathematical models with very complicated dynamics." Nature, Vol. 261, June 10, 1976, pp. 459-467; reprinted in The Theory of Chaotic Attractors, pp. 85-93. Springer, New York, NY, 2004. The sequences listed in Table 2 are A000079, A027375, A000031, A001037, A000048, A051841. - N. J. A. Sloane, Mar 17 2019
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LINKS
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FORMULA
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a(n) = 1/(2*n)*Sum_{d|n} gcd(d,2)*mu(d)*2^(n/d).
a(n) = 1/(2*n)*Sum_{k=1..n} gcd(gcd(n,k),2)*mu(gcd(n,k))*2^(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = (1/n)*Sum_{k=1..n} gcd(n/gcd(n,k),2)*mu(n/gcd(n,k))*2^gcd(n,k)/phi(n/gcd(n,k)). (End)
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EXAMPLE
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a(5) = 3 = |{ 00011, 00101, 01111 }|.
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MATHEMATICA
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a[n_] := Sum[GCD[d, 2]*MoebiusMu[d]*2^(n/d), {d, Divisors[n]}]/(2n);
Table[a[n], {n, 1, 32}]
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PROG
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(PARI)
L(n, k) = sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d) );
a(n) = sum(k=0, n, if( (n+k)%2==0, L(n, k), 0 ) ) / n;
vector(33, n, a(n))
(Haskell)
a051841 n = (sum $ zipWith (\u v -> gcd 2 u * a008683 u * 2 ^ v)
ds $ reverse ds) `div` (2 * n) where ds = a027750_row n
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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