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A042981
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Number of degree-n irreducible polynomials over GF(2) with trace = 1 and subtrace = 0.
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6
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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
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OFFSET
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1,6
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LINKS
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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
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FORMULA
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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)}.
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Frank Ruskey
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STATUS
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approved
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