OFFSET
1,5
COMMENTS
Also number of trace 0 irreducible polynomials over GF(2).
Also number of trace 0 Lyndon words over GF(2).
REFERENCES
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
LINKS
FORMULA
a(n) = 1/(2*n)*Sum_{d|n} gcd(d,2)*mu(d)*2^(n/d).
a(n) ~ 2^(n-1) / n. - Vaclav Kotesovec, May 31 2019
From Richard L. Ollerton, May 10 2021: (Start)
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)
EXAMPLE
a(5) = 3 = |{ 00011, 00101, 01111 }|.
MATHEMATICA
a[n_] := Sum[GCD[d, 2]*MoebiusMu[d]*2^(n/d), {d, Divisors[n]}]/(2n);
Table[a[n], {n, 1, 32}]
(* Jean-François Alcover, May 14 2012, from formula *)
PROG
(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))
/* Joerg Arndt, Jun 28 2012 */
(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
-- Reinhard Zumkeller, Mar 17 2013
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Frank Ruskey, Dec 13 1999
STATUS
approved