login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A042979 Number of degree-n irreducible polynomials over GF(2) with trace = 0 and subtrace = 1. 11
0, 0, 1, 0, 2, 2, 4, 8, 13, 24, 48, 80, 160, 288, 541, 1024, 1920, 3626, 6912, 13056, 24989, 47616, 91136, 174760, 335462, 645120, 1242904, 2396160, 4628480, 8947294, 17317888, 33554432, 65074253, 126320640, 245428574, 477211280, 928645120, 1808400384, 3524068955, 6871947672, 13408665600, 26178823218 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

K. Cattell, C. R. Miers, F. Ruskey, J. Sawada and M. Serra, The Number of Irreducible Polynomials over GF(2) with Given Trace and Subtrace, J. Comb. Math. and Comb. Comp., 47 (2003) 31-64.

F. Ruskey, Number of irreducible polynomials over GF(2) with given trace and subtrace

FORMULA

a(n) = (1/n) * Sum_{k=0..n, n+k == 0 (mod 4)} L(n, k), where L(n, k) = Sum_{d|gcd(n, k)} mu(d)*binomial(n/d, k/d).

MATHEMATICA

L[n_, k_] := Sum[ MoebiusMu[d]*Binomial[n/d, k/d], {d, Divisors[GCD[n, k]]}]/n; a[n_] := Sum[ If[ Mod[n+k, 4] == 0, L[n, k], 0], {k, 0, n}]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Jun 28 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)%4==0, L(n, k), 0 ) ) / n;

vector(33, n, a(n))

/* Joerg Arndt, Jun 28 2012 */

CROSSREFS

Cf. A042980, A042981, A042982.

Cf. A074027, A074028, A074029, A074030.

Sequence in context: A104700 A039941 A036761 * A000018 A357307 A306604

Adjacent sequences: A042976 A042977 A042978 * A042980 A042981 A042982

KEYWORD

nonn

AUTHOR

Frank Ruskey

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 5 18:47 EST 2023. Contains 360087 sequences. (Running on oeis4.)