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A042981 Number of degree-n irreducible polynomials over GF(2) with trace = 1 and subtrace = 0. 6

%I

%S 1,0,1,1,1,3,4,8,15,24,48,85,155,297,541,1024,1935,3626,6912,13107,

%T 24940,47709,91136,174760,335626,645120,1242904,2396745,4627915,

%U 8948385,17317888,33554432,65076240,126320640,245428574,477218560,928638035,1808414181,3524068955,6871947672,13408691175,26178823218

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

%H Vincenzo Librandi, <a href="/A042981/b042981.txt">Table of n, a(n) for n = 1..1000</a>

%H K. Cattell, C. R. Miers, F. Ruskey, J. Sawada and M. Serra, <a href="https://www.researchgate.net/publication/2634456_The_Number_of_Irreducible_Polynomials_over_GF2_with_Given_Trace_and_Subtrace">The Number of Irreducible Polynomials over GF(2) with Given Trace and Subtrace</a>, J. Comb. Math. and Comb. Comp., 47 (2003) 31-64.

%H F. Ruskey, <a href="http://combos.org/TSpoly">Number of irreducible polynomials over GF(2) with given trace and subtrace</a>

%F 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)}.

%t L[n_, k_] := Sum[ MoebiusMu[d]*Binomial[n/d, k/d], {d, Divisors[GCD[n, k]]}]/n;

%t a[n_] := Sum[ If[ Mod[n+k, 4] == 1, L[n, k], 0], {k, 0, n}];

%t Table[a[n], {n, 1, 32}]

%t (* _Jean-Fran├žois Alcover_, Jun 28 2012, from formula *)

%o (PARI)

%o L(n, k) = sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d, k/d) );

%o a(n) = sum(k=0, n, if( (n+k)%4==1, L(n, k), 0 ) ) / n;

%o vector(33,n,a(n))

%o /* _Joerg Arndt_, Jun 28 2012 */

%Y Cf. A042979, A042980, A042982.

%Y Cf. A074027, A074028, A074029, A074030.

%K nonn

%O 1,6

%A _Frank Ruskey_

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Last modified August 13 08:27 EDT 2020. Contains 336442 sequences. (Running on oeis4.)