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A042981 Number of degree-n irreducible polynomials over GF(2) with trace = 1 and subtrace = 0. 3
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 (list; graph; refs; listen; history; text; internal format)
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-A042982.

Cf. A074027-A074030.

Sequence in context: A104370 A310013 A033854 * A007486 A027977 A165438

Adjacent sequences:  A042978 A042979 A042980 * A042982 A042983 A042984

KEYWORD

nonn

AUTHOR

Frank Ruskey

STATUS

approved

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Last modified January 15 19:33 EST 2019. Contains 319171 sequences. (Running on oeis4.)