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A042982 Number of degree-n irreducible polynomials over GF(2) with trace = 1 and subtrace = 1. 7
0, 1, 0, 1, 2, 2, 5, 8, 13, 27, 45, 85, 160, 288, 550, 1024, 1920, 3654, 6885, 13107, 24989, 47616, 91225, 174760, 335462, 645435, 1242600, 2396745, 4628480, 8947294, 17318945, 33554432, 65074253, 126324495, 245424829, 477218560, 928645120, 1808400384, 3524082400, 6871947672, 13408665600, 26178873147 (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_{ L(n, k) : n+k = 3 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] == 3, L[n, k], 0], {k, 0, n}]; Table[a[n], {n, 1, 32}] (* Jean-Fran├žois 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==3, 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: A056224 A293674 A052527 * A006367 A246807 A077902

Adjacent sequences:  A042979 A042980 A042981 * A042983 A042984 A042985

KEYWORD

nonn

AUTHOR

Frank Ruskey

STATUS

approved

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Last modified January 15 18:43 EST 2019. Contains 319162 sequences. (Running on oeis4.)