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A051841 Number of binary Lyndon words with an even number of 1's. 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Also number of trace 0 irreducible polynomials over GF(2).

Also number of trace 0 Lyndon words over GF(2).

LINKS

T. D. Noe, Table of n, a(n) for n=1..300

Index entries for sequences related to Lyndon words

F. Ruskey, Number of q-ary Lyndon words with given trace mod q

F. Ruskey, Number of Lyndon words of given trace

FORMULA

a(n) = ( Sum_{d|n} gcd(d, 2)*mu(d)*2^{n/d}) / (2n).

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

Same as A001037 - A000048. Same as A042980 + A042979.

Cf. A027750, A008683.

Sequence in context: A002823 A109509 A006053 * A096081 A054162 A174783

Adjacent sequences:  A051838 A051839 A051840 * A051842 A051843 A051844

KEYWORD

nonn,easy,nice

AUTHOR

Frank Ruskey, Dec 13 1999

STATUS

approved

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Last modified January 15 20:47 EST 2019. Contains 319184 sequences. (Running on oeis4.)