A042981 Number of degree-n irreducible polynomials over GF(2) with trace = 1 and subtrace = 0.

1, 0, 1, 1, 1, 3, 4, 8, 15, 24, 48, 85, 155, 297, 541, 1024, 1935, 3626, 6912, 13107, 24940, 47709, 91136, 174760, 335626, 645120, 1242904, 2396745, 4627915, 8948385, 17317888, 33554432, 65076240, 126320640, 245428574, 477218560, 928638035, 1808414181, 3524068955, 6871947672, 13408691175, 26178823218

Offset

1,6

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

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] == 1, 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==1, L(n, k), 0 ) ) / n;
vector(33, n, a(n))
/* Joerg Arndt, Jun 28 2012 */

Crossrefs

Cf. A042979, A042980, A042982.

Cf. A074027, A074028, A074029, A074030.

Sequence in context: A104370 A310013 A033854 * A337438 A007486 A027977

Adjacent sequences: A042978 A042979 A042980 * A042982 A042983 A042984

Keyword

nonn

Author

Frank Ruskey

Status

approved